Leftovers are better, and here's why.
Even for the sweeper, shell bell can only give back 1/8 of the damage given.
However, the ideal sweeper is fast and strong, allowing it to OHKO without being stricken back.
Furthermore, the shell bell doesn't allow for stat modifying moves like calm mind or swords dance,
because it won't replenish HP on these turns.
And if one is to Baton Pass stat modifiers onto a sweeper, even if it be, say, two stages attack and one
of speed, than these increases (which are bigger than they seem) should allow for OHKO's frequently.
So one can conclude that, although neither leftovers or shell bell are good items for sweepers, shell bell is probably worse.
It's better to keep a sweeper holding something that allows it to do more damage, not take more damage.

For tanks and sponges, who are designed to take some hits and not always strike back, leftovers are clearly the better choice.

For shufflers, primarily Skarmory (with Spikes/Toxic and Roar), leftovers are probably STILL better because shell bell doesn't restore
health based on indirectly dealt damage. Correct me if I'm wrong here.

Shell bell+OHKO moves... well, OHKO moves are pretty terrible to depend on for anything, with 30 accuracy... and sturdy trumps them, too.

The above situations are pretty broad, but I can think of a few exceptions where shell bell might be better:
1) Raticate's Super Fang (if it were traded from FRLG. It uses SF, and automatically gains 1/16 of its opponents HP at the time. Great if battling something with ridiculous HP)

2) Anything's Seismic Toss/Night Shade (at level 100, especially when used by something like Dusclops, which, despite great defenses, has terrible HP and 25 guaranteed HP is more than 1/16 of its own HP every turn)

3) Counter (maybe on Aggron with it's 4x weaknesses to Fighting and Ground, or Blissey, but only if this Counter/shell bell business works)

(Sorry if this is posting later than the allowed time frame, but I really want to get this down and said FINALLY)
(also, sorry this is so LONG! But, it was SO MUCH FUN putting together! >_<;

Since NO ONE has been willing to do so (or even taken the time to think of it) here's some REAL help on the subject.
(for those who want to avoid all the hassle of my VERY BAD example layout, just skip to the bottom to see the REAL ANSWER TO THIS QUESTION)

I DO have one question myself though, and it really does apply to real practices of this: does the Shell Bell apply to the TOTAL damage dealt, or does it only calculate up to the KO and the rest is moot? (OBVIOUSLY, you can't gain more than your own max HP)
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Situation:
Lv. 100 [pokemon]
HP: 400 (Max IV's/EV's, just for kicks; AND that's a high max HP for most species, so don't expect this example's Leftovers to be as good for most Lv. 100's, though there are a fair few that are that good)
Speed: (this is just an example, so the number doesn't matter here, but assume it's faster than what it's gonna face)
Attack/Special Attack: (this is just an example, so the number doesn't matter here, but assume it's quite strong)
Attacks: Seismic Toss/Nightshade, [some powerful attack], [some multi-hit attack]
Item: Leftovers (1/16 of holders max HP, every turn)/Shell Bell (1/8 of DAMAGE DEALT TO OPPONENT with EACH ATTACK)

Opponent: (come up with it yourself)

Leftovers situation:
-Leftovers gains ALWAYS 25 HP in this case. Period.

Shell Bell situations:
-After using Seismic Toss/Nightshade:
--12.5 HP
-After using [some powerful attack] hitting type-neutral, no STAB, you did 150 damage:
--18.75 HP
-After using [some multi-hit attack] that hit's 2-5 times, hitting type-neutral, no STAB, doing from 50-125 damage:
--6 HP-16 HP

BUT, what if you were smart and instead had several powerful attacks so you can pretty much always get STAB and/or super-effective?
Using the earlier number of 150 damage:
+STAB it did 210 damage
--26 HP
+x2 Super-effective it did 300 damage
--37 HP
+Stab+x2 Super-effective it did 400 damage
--50 HP
+x4 Super-effective it did 600 damage
--75 HP
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So, while YES I KNOW there are tons of general problems with this example, (more than I think I realize) it all boils down to this:

If you've got 400-714 (MAX) HP and aren't a sweeper: Leftovers for 25-45 HP. (The MAXIMUM is 45, and only if it's a Lv. 100 Blissey with fully maximum HP)
-Almost EVERYTHING that breaks 450 HP is NOT likely to be a major competitor, unless they are Support or the match allows Legends/Uber.
--This means that everything ELSE with Leftovers isn't capable of even getting 28 HP out of it, period.
---300 HP marks 18 HP

If you've got a sweeper with ANY amount of HP, and is capable of averaging 200 damage in most situations, they will get at least 25 HP each attack.
-Yeah, that's a pretty big amount of damage you're gonna have to do to get that much, but you have to actually compare that to what you'd get from Leftovers to really see the difference.
--Also, super-effective or even just STAB is pretty commonly desired, so keep in mind those multipliers.
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In shorter, THESE ARE THE EXACT NUMBERS TO BASE YOUR USAGE ON:

To regain 1 HP per turn (minimum, except for Shedinja who always gains 0 HP):
-Leftovers: have 16+ HP
-Shell Bell: do 8+ damage each turn

To regain 5 HP per turn:
-Leftovers: have 80+ HP
-Shell Bell: do 40+ damage each turn

To regain 10 HP per turn:
-Leftovers: have 160+ HP
-Shell Bell: do 80+ damage each turn

To regain 15 HP per turn:
-Leftovers: have 240+ HP
-Shell Bell: do 180+ damage each turn

To regain 20 HP per turn:
-Leftovers: have 320+ HP
-Shell Bell: do 160+ damage each turn

To regain 25 HP per turn:
-Leftovers: have 400+ HP (78 Pokemon; 24 are Legendary)
-Shell Bell: do 200+ damage each turn

To regain 30 HP per turn:
-Leftovers: have 480+ HP (12 Pokemon)
-Shell Bell: do 240+ damage each turn

To regain 35 HP per turn:
-Leftovers: have 560+ HP (3 Pokemon: Wobbuffet, Chansey, & Blissey)
-Shell Bell: do 280+ damage each turn

To regain 40 HP per turn:
-Leftovers: have 640+ HP (2 Pokemon: Chansey & Blissey)
-Shell Bell: do 320+ damage each turn

To regain 45 HP per turn:
-Leftovers: have 720 HP (the MAXIMUM is a maxed-out Blissey, which has 714 HP; so she'd get 44-45 HP)
-Shell Bell: do 360+ damage each turn

To regain 50 HP per turn:
-Leftovers: IT'S IMPOSSIBLE, as no Pokemon can legally HAVE that high of HP (which would be 800 HP, if you're curious)
-Shell Bell: do 400 damage each turn

To regain 90 HP in one turn:
-Leftovers: IT'S IMPOSSIBLE, as no Pokemon can legally HAVE that high of HP (which would be 1440 HP, if you're curious)
-Shell Bell: do 720 damage. (ie: Knock out a max-HP Blissey in one shot. It's FAR from impossible, and can be quite easy if you use the right Pokemon, but it's only gonna happen once in a match; unless they revive it with the Miracle Shooter.)

To regain 4,096 HP in one turn:
-Leftovers: have 65,535 HP (CHEATER CHEATER CHEATER!!! Apparently, the game's data is able to handle a technical maximum of 65,535 for a stat... but that's obviously cheating. [65,535 is FFFF in Hexadecimal, by the way. It's just like how in the Gen I & Gen II games it could only calculate up to FF, or 255) So, this is THE maximum for Leftovers.
-Shell Bell: do 32,768 damage each turn (VERY unlikely... but NOT impossible, and you DON'T have to cheat to do it O_O; )

To regain 1,267,815 HP in one turn:
-Leftovers: Game's data restricts HP stat to FFFF (65,535) so this is impossible for Leftovers.
-Shell Bell: do 10,142,520 damage (see my following example of how to LEGALLY do this O_O; I seriously freaked out when I realized what this means about Espeon and Gardevoir...)
-A way to LEGALLY do this (unless I messed up my math somewhere, I'd be happy if someone checked): Max IV/EV Espeon (130 Sp. Attack) using a maximum-Baton Passed-powered-up (Sp. Attack x4) Assist Power {860 base}, holding Choice Specs (Sp. Attack x1.5), powered by Helping Hand (x1.5), and against a Lv. 1, 0-all IV/EV Croagunk (Super Effective x4), Naive-nature (4 Sp. Def.), that has been hit by Acid enough to minimize it's Sp. Defence (Sp. Defence x0.25) and dealing a Critical Hit (Crit. x2)= (using Smogon's calculations)
(((((((100 × 2 ÷ 5) + 2) × 1290 × 780 ÷ 50) ÷ 1) × 1) + 2) × 2 × 1 × 100 ÷ 100) × 1.5 × 2 × 2 × 1)=
10,142,520 damage)

Is that epic or what?

NOW tell me Leftovers is ALWAYS better than Shell Bell! HA! (not that I actually knew the comparison until today... -_-;)

Essentially: If you expect to have a Pokemon do around 50% or more damage than it's OWN maximum HP, you want Shell Bell. Otherwise, Leftovers.
(If [damage] > [HP/2]=Shell Bell; otherwise=Leftovers)

wow

I think I will have to read that a couple more times before I can rap my head around it.

^_^; Hope it helps! Took me a while of reading just to partially keep the math right in my head long enough to get that typed up. Needless to say, I'd have to re-read that to do so again. -_-;

Sorry it's not very well-written. I certainly wouldn't mind if someone were to make a shorter version that is "as clear". (Especially if I did make some math mistakes; that wouldn't surprise me, since even though I'm fairly good at math, I tend to easily make mistakes and have to double back.)

The only thing that really matters out of that whole thing is the actual comparison of the ratios required to choose which one to use. -_-;

Those are great examples, but the problem is that this is about 3rd gen. So yeah, Espeon has epic Stored Power with specs, but unfortunately that's in gen 5.

In general, sweepers should be holding Lefties or Lum Berry. A full-on powerhouse sweeper might prefer choice band, and a subsitute/endure sweeper will want a pinch berry. Otherwise, Leftovers or Lum Berry.

Shell Bell's best use this gen is for level 1 Aron in sandstorm. Aron survives one hit thanks to sturdy's new mechanics, then uses Endeavor to bring them down to very little damage. Sandstorm finishes them off. Aron then recovers back to full health from Shell Bell, and the process repeats. And yeah, ghost types take a dump on this strategy.

If you're looking for the most powerful attack possible, I think it's Shuckle. A power swapped, defence curled final rollout with all the other unlikely boosts and stuff is the most powerful possible attack in the game.

O_O; ... I hadn't noticed this thread was for 3rd gen. -_-
Sorry to any offended by my lack of notice.

Ah well. Still, the math still stands, regardless of generation, so it still applies to 3rd Gen too.

By the way, here's the page on Smogon with the formula (it's for fourth-gen, which I'm pretty sure accurately applies to 3rd and 5th, just with a couple item/attack tweaks):
http://www.smogon.com/dp/articles/damage_formula

Yeah, I know about F.E.A.R. Aron, but it's not really worth mentioning. Just hit it with Will-O-Wisp and it's no longer sweeping. Not that that's necessarily easy, but neither is it smart to rely on that Aron.
Anyway.

Shuckle's Rollout damage:

Rollout's max. power is 960 (with Defense Curl).
Shuckle's Swapped Attack would be 614.

The Choice-item will have to have been Swapped onto them, which means it's absolutely impossible in single battles, as far as any scenario I can think of goes.

53,474,020.8 (round off, and some possible slight calculation variances could alter the damage 1 or 2 points.)

So, yeah, that's about 5 times more damage (assuming the same kind of extreme situation, where the foe is literally the ultimate lv. 1 shrimp with min. defence possible and has no assistance to protect it.) than the Stored Power Espeon.
This would "regain" 427,792,160 HP with Shell Bell.

The main difference, obviously, is how much more unlikely it is. Stored Power Espeon is a lot more feasible, since it's just a case of (VERY unlikely) max-boostings getting copied/Passed to the Espeon. But it's still more feasible, since it's gonna be having those defensive stat boosts as well. Plus, it's still hugely powerful even with just Calm Mind boosting (260 base power with 6 Calm Minds, stronger than Explosion, and self-powered), plus any it can get from others. There's still a TON of ways to stop it, but those are still better odds than this:

Shuckle... is gonna be out of the running before it can get to the 5th Rollout, no matter what you're doing. (heck, it'll be out before the first).
-Burn. At any point, if it's Burned, this whole plot involves half the damage. Technically, the final Rollout would still be bigger than Espeon's Stored Power...
-Damage of any kind. The MOMENT it uses Power Swap it's gonna be KO'd instantly, and won't get a single Rollout going. (Unless you use Trick Room; then you get one attack before the KO. -_-; ).
-Luck. Assuming that you manage to get to using the boosted Rollout, you've got to hope it even HIT's five times to get that big number. Not only can they force you to miss/fail to reset the power, but it can do that to itself. (plus I did account for Critical Hit, so that's another luck boost; of course, that does apply to the Espeon example as well.)

Another note: Ice Ball Walrein can do this same combination, but it has about 1/3 the base Attack to start off with. I'm not curious enough to check the damage output in that case, though.

WOW! That was fun to check! O_O Cripes, that's a LOT of damage... Kinda makes me want to see what it does against the biggest defending Lv. 100 (something)... but I'm not going to.

It's a 3HKO on max defenses, +6 Steelix. All this Shuckle stuff was done and tried back in gen 3

^_^ I'm not really interested in the X-hit KO counts. I just like seeing the actual damage output. It is one of those (VERY FEW) things that has always slightly bugged me about Pokemon: you NEVER (outside of Shedinja) know how many HP your opponent has! I'm generally good, but I just can't bring myself to seriously try to remember the general HP range of ALL high-use Pokemon (especially now that there is a total of over 600...). Anyway, because of that I've always wished I could see just how much damage I'm actually doing.
Actually... that is a definite use of the Shell Bell in battles... you could approximate how much damage you're doing and estimate your opponent's HP! Just multiply the HP you recover by 8 and you know about how much damage you did! Hm, I never thought of that. For some people, could be a way to scout a Pokemon's defenses. (Not that I seriously think it would be any much use in high-end matches; still, could be neat!)

maximum at level 100, base 100 HP is 404. That's a fact you should remember. Then just approximate around that... So base 50 is gonna be about 250, and base 150 is gonna be about 550.

overall Leftovers is more dependent because it agree with a huge chunk of pokemon while shell bell works only with a specific group of pokemon that can dish out major damage, especially pokemon with Choice Band such as Machamp, Deoxys Atk, Aerodactyl, etc.

Sigh...
Not really. Tons of Pokemon would be "better off" with Leftovers than Shell Bell, but lots of them are even better off with something else altogether. That's what I was showing.

It's ALL circumstantial, the is NO single all-encompassing true answer. This is the closest I can figure to one (mainly by pointing out when you would not want to use them):

-If your HP is not high enough, Leftovers is most likely a waste, since you'll gain too few HP to make a difference; unless you have a strategy that works with that setup.

-If you won't do high enough damage (relative to your HP), Shell Bell is most likely no better than Leftovers and thus would be switched for it unless your HP are too low, in which a totally different item would be better; unless you have a strategy that works with that setup.

-If you:
--don't do enough damage (relative to your HP) (recovery = 1/8th damage dealt)
--don't have high enough HP (recovery = 1/16th your maximum HP)
--don't have a strategy that would make use of the HP-recovery despite HP/damage-output not being the usually desirable numbers
then: Use something else.

Honestly, despite how many people try to make it sound, most Pokemon are much better off without the HP-recovery items. Sure, Leftovers CAN work with anything, but that doesn't mean it's a good choice. Remember my calculations earlier? If you want to recover enough for it to matter, you generally need at least 320 [minimum base HP: 58, at Lv. 100] (+20HP/turn). And that's only if you invest for MAXIMUM HP, which most Pokemon would rather trade for higher Attack/Sp.Attack/Speed.

Also, you have to take into account that often the standard battle has all Pokemon reduced to Lv. 50, which means that only two Pokemon are even capable of having that much HP. Of course, in Lv. 50 matches offensive/defensive stats are reduced as well, so the bar is lower as well as to how much is needed to recover to make a difference. However, being at lower stats means that attacks that have a higher base damage are going to hurt a little more than in the same situation at Lv. 100.

It's all in how they are focused. Really, I need to stop saying anything on this, since I've said the same thing several times... Sorry if I've bothered anyone with my comments. -_-;

Hopefully, people who are interested will be able to read through the junk text and find what it is that actually matters regarding this topic.

Also, I'm going to hope that a MUCH more useful and condensed page that includes the core info of this topic appears on Bulbapedia so that it's easy to find for anyone interested.

Really, I don't think that there is anything more to say about this topic that isn't just opinions on what a given person thinks.

Leftovers always increases 1/16th of your health per turn. So in proportion to max HP, every Pokemon recovers about the same amount of health with leftovers. A Shuckle has about 150 HP, whereas a Blissey has about 700. Sure Blissey will recover more health from leftovers, but Shuckle will take much, much less damage than Blissey. What does 350 HP damage to Blissey might do 75 HP damage to Shuckle (not necessarily true, but the principle's the same). So although Shuckle recovers less HP from leftovers than Blissey does, in comparison to max HP and the amount of health they will taking, they recover the same.

Obviously this depends on how bulky the Pokemon is. Giving leftovers to Blaziken is just a waste.

Also, this is gen 3 we're talking about, so a huge majority of the time Leftovers is the preferred item. This includes slightly bulkier setup sweepers, such as Salamence and Dragonite, almost all walls, like Skarmory and Blissey, all subsitute users, like Sceptile and Ninjask, and all anti-metagame pokes, like Claydol and Celebi. This pretty makes up the majority of Pokemon.

I think leftovers are better because you are healing for sure

I like Leftovers more, even though Leftover healing is mediocre.