The 3 classes in the Subject line are in our party. The group is curious to know who is the hardest hitter of the 3, and who is the most reliable. The Barbarian is of the Dragon Order, the Druid is of the Wild Order, the Rogue is of the Theif racket.
Without doing lots of math and plots, what I think:
The barbarian and rogue are both using their main stat for attack and damage. They get proficiency at the same levels. So they should have very similar accuracy. The rogue can apply flatfooted w/o a flank, so there might be times they can be more accurate, but hopefully you have a pair flanking.
The druid will lag, their attack stat is not main stat, so it will be at least one behind. When wild-shaped (which don't really work until level 3), they get a plus 2, so at level 3 they'll beat the other two, before they get expert. from 3-> 11 they'll be -1/0 before depending on ability boosts. So when wild-shaped they stay competitive. When not wild-shaped, if they're using attack spells off WIS, they can be comparable accuracy to the martial.
Looking at expected damage:
The barbarian and rogue will have equal ability modifiers to damage. The barbarian likely have a higher damage die weapon. The barbarian also gets a flat +4 from rage(at low levels) on top of that. The rogue has a smaller damage die weapon, plus sneak attack, but sneak attack is on average less damage then the rage bonus. The rogue is rolling more dice though, especially as sneak attack increases. The rogue should be slightly more 'average' in damage due to the number of dice being rolled. (basically almost always double the number of dice the barbarian has, assuming both are keeping their striking runes up to date)
The wild-shaped druid mainly falls behind both due to the lowest flat plus (+1 for the first animal form, compared to +4s that the others have, but will typically have 2 competitive damage dice (between the rogue/barbarian). The higher level shapes often get a high flat plus, or better dice, but almost never both in a way that's competitive -- but it is more flexible on weapon types/reach requirements for any particular fight.
Without doing lots of math and plots...
And to do the math to answer the question we would need to know all the stats and bonuses involved.
But here is a quick math crash course in order to calculate it:
First you need to know the probability of successful attack. So compare the attack bonus of the attack to the defense bonus of the defender. Determine the count of the numbers of the d20 that will succeed (example: success on a roll of 14 results in a number count of 7). Multiply that count by 0.05 to get the probability of success (this is ignoring critical hits, but it will still be useful for comparisons between the characters).
That by itself will tell you which character is more reliable in doing damage - the character with the highest probability value.
For expected damage, you first need to calculate the expected value of the dice being rolled. For each die, that value ends up being half of the highest value possible plus 0.5. So a d6 would be 3.5, a d8 would be 4.5, a d12 would be 6.5 and so on. Add up all the expected values of the dice being rolled for the attack and add in any flat bonuses. That is your expected damage on a hit.
From there, multiply the expected damage with the probability of success to get the expected damage for an attack.
Then you can calculate that for each attack of every character in the party at every MAP level in order to fully answer the question.
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A quick rundown at at level 6.
And each have 18 or 19 in Str/Dex.
Everyone has 14+stuff to hit. So they all hit the same.
Damage
Druid: 2d8+5 = 14
Barbarian: 1d12+4+4 = 14.5
Rogue: 1d6+4 = 7.5
Sneak attack: 3d6+4 = 14.5
Pretty even all around. But I would probably say Barbarian would edge out. They have stuff like sudden charge, making it easier to make 2 attacks.
Rogue second, as they will often need to spend an action to get sneak attack. Though that is partially made up by having easier flat-footed.
Druid last, as he has no combat feats. Though, he can cast fireball before wild-shaping, making up quite a bit of difference. Or maybe cast haste on the rogue.
Just ran the full math, excluding magic items (which makes a difference) for accuracy.
Rogue and Barbarian are equal to-hit at each level.
Wild Druid lags by -1 at level 1&2 (no wild shape, and thus no extra bonus, Lower ability mod, equal proficiency)
Wild Druid is +1 to hit at 3&4 (wild shape bonus, lower ability, equal prof)
Wild Druids are equal from 5-9 (wild shape bonus vs high prof, equal ability mod)
Wild Druid lags by -1 at 10. (Second boost, martials at 19)
Wild Druid leads by +1 at 11&12. (All at expert now)
Wild Druid lags at 13/14 (Martials at Master)
Wild druid equals at 15-19, (druid catches up with ability mod)
Wild druid lags by -1 at 20 (ability mod increase for martials)
I haven't been following the mighty handwrap discussion, but the above analysis would lead me to be predisposed to accepting wild shaped druids benefit from the to-hit aspect at least.
Figuring out the damage side will take a bit more time. But as Mellored, indicates (and as tivadar's thread suggests), this kind of spreadsheet only analysis misses a LOT of dynamics. The Barb with one feat investment (sudden charge) is likely to always be able to contribute in the first round (rage + sudden charge == got their self buffs, and mobility, likely getting one full strength attack off), Rogue no feats, has double move + attack or move, feint, attack as options, one of which is likely to be applicable (one at reduced damage though). Druid, spending almost all their class feats (to keep up with damage/ac) 2 actions to transform first round, no attack if needing to move. Can't combo a blasting spell with a shape change, so basically losing a round (could shoot a missile weapon, but that's going to be at poor accuracy, no +2, and using a tertiary stat at best)
Ran most of the damage stuff, this includes striking runs for the Barbarian/Rogue and not for the druid.
Pretty sure there can be some minor errors, but the general look is (for first attack only), the barbarian has a slight lead on the rogue. And the druid is about one barbarian damage die behind for most of the time. Levels 14 and 18, things align just right that basically all three are the same.
Rogues will be slightly more accurate than the barbarian on second/third attacks, (due to agile), so that will let them close the gap on turns they can attack more than once. Druids won't benefit as much, as often the forms that have agile, will also have a lower first attack than used for the analysis, so they're further behind.
Note: regrading handwraps here, it does NOT look like any form of consistent interpretation of what adding a damage die to all the wild-shapes makes the damage difference any more neutral/balanced -- mainly because the difference between the martials and the druid does not increase at the levels where the martials upgrade their striking runes.
The druids one saving grace on the damage front is the ability to customize damage types (including energy types at some tiers) to the battle at hand. But if a martial has a shifting rune, that lets them cover the physical types, in a more action efficient version.