Obviously the answer is "depends" but I want to unpack that depends a bit more.
Assuming the to hit bonus and damage bonus are falling in a similar range of value to each other, rather than being skewed to one extream or the other, how much more important is the to hit bonus? Statistically is it twice as valuable, a third, a tenth? What do those with the power of math have to say?
Your topics a little messed up? I think you meant to ask how much more valuable is "to hit" than "Damage" A to hit and attack bonus are the same thing.
The math on this varies wildly depending on the number of attacks you have and the damage per attack. If you have one , huge, monstrous swing with a flaming shocking aciding freezing greatsword then another 2 points of damage gets you little, while + 1 hit gets you a fair bit more DPR.
The damage formula is h(d+s)+tchd.
h = Chance to hit, expressed as a percentage
d = Damage per hit. Average damage is assumed.
s = Precision damage per hit (or other damage that isn't multiplied on a crit). Average damage is again assumed.
t = Chance to roll a critical threat, expressed as a percentage.
c = Critical hit bonus damage. x2 = 1, x3 = 2, x4 = 3.
you get an attack bonus for the cost of random 0-5 elemental damage.
but honestly, it depends.
It depends on your attack and ennemy defenses.
Take your standard attack, and an ennemy with standard in combat AC.
If I had to make something up, I'd say 1 attack is (AC-ATK-1) times more important than 1 damage.
Yeah, it really depends on what your trying to hit (AC), and how much damage you are doing already.
If you aren't guaranteed to hit (or miss) the target (well 5% chance), such that a +1 won't affect the roll, then the damage is obviously better.
If the +1 to attack would improve your chances by one on the d20 roll, that's increasing ALL damage you do by 5%.
If you roll 1d4, then a +1 damage will probably improve things better (for the most part).
If you have 1d8+20, then a 5% improvement is over 1 point of damage already.
And that's not taking into account how attack bonus can affect iteratives, or retaining good chance to hit allowing you to use things like Power Attack or Deadly Aim more effectively.
And since you can't know what AC you are going to come up against (and with things like fighting defensively, higher ground, spell buffs, etc, it can change mid-combat), a fairly high attack roll is typically pretty important.
Without bothering with the overly complicated math equations... I think it boils down to this.
If you pick damage bonus... you hurt more.
If you never HIT the target, then it doesn't matter HOW high your damage is.
By this philosophy I'll pick 'to hit' over damage any day :P
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one of the stats on the DPR olympics threads are how much damage goes up by per point of hit. Its not uncommon to see 2. something there. Find your build, find something similar in the thread, and it should give you an idea.
If you needed a 19 to hit, a +1 to hit will increase your average damage per attack by 50%. If you needed a 3 to hit, a +1 to hit will increase your average damage per attack by about 5.5%.
Power attack allows you to trade 1 accuracy for 2 damage (3 with a two-handed weapon). This is often a good trade, but not against high AC.
If you wanted a simple and clear answer: +1 to hit is worth, let's say... +2.5 damage.