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Most weapons of the same level and rarity have pretty identical attack stats (at least in shops anyway), but the actual damage they deal is drastically different. Example: a 100 attack hand cannon deals a lot more damage than a 100 attack auto rifle.
Is there a formula or way to tell how much damage I can actually expect to be dealing with this gun when it has that attack?
So far my data points to this being the right explanation: Impact is a measure of how much to scale the Attack of the weapon by, to determine a per-bullet damage. If you make a plot betwen Impact and bullet damage in weapons with the same Attack (sidenote: all weapons of the same level, rarity and weapon slot have the same Attack) you'll see a correlation. I don't know the exact scaling factors and numbers involved, or whether this changes as you go up towards legendaries/exotics as my data is mostly from the most consistent baseline source of weapons I have (white primary engrams from Cryptarch, which give lvl 15 weapons). I also only own a single legendary and no exotic weapons.
This means that if you're comparing Attack directly you might as well be comparing levels. Impact will give you a rough idea of how large the single-shot damage will be relative to other weapons with the same Attack, but if you're comparing a lower Attack weapon with one with a higher Attack, you'll have to do some quick headmath to determine if the weapon will do the right kind of damage for your usage pattern, based on impact difference and rate of fire. In general at this point it's better to just testfire the weapon.
The data I refer to is available in this google spreadsheet
I can only offer educated guesses on my experience while playing the game, and short of a developer jumping on here an answering flat out I doubt we will ever have a definitive answer.
I believe the damage value is the exact same representation that you would see on weapons in Diablo 3, it is an aggregated value based on the stats of the weapon. In the case of destiny it would be a calculation of damage per round then normalized/averaged over a defined time period in this case likely a period of 1 sec.
I am firm in this belief because it is the only explanation I can come up with for the differences you note between a hand cannon and an auto rifle.
TL;DR; The damage value is Damage per second and not Damage per shot.
Impact stat is how much damage you deal per shot. The higher the impact, the more damage you deal each time a bullet connects. High impact weapons also tend to have low rates of fire.
The attack stat is a multiplier that affects how much damage you deal to different level enemies. It's fairly complicated. (It helps to think of it as a kind of armour pierce)
Basically, in PvP, ignore the attack stat completely, unless you're in a mode that has level advantage enabled.
For PvE, just try out weapons. High attack weapons deal more damage to higher level enemies than low attack weapons, but you just need to find a balance.
Impact does determine dmg per shot.
The rate of fire/impact stats of weapons can be used to compare similair weapons of the same type. e.g. a lvl 20 sniper with high impact will deal more dmg per shot than a lvl 20 sniper with lower impact, but also fire slower.
Also, don't try to compare the impact and rate of fire stats of different weapon types. (A pulse rifle can have an apparent higher impact than a sniper rifle, but will not deal more dmg per shot than the sniper. Same for rate of fire. These values are just for comparing sniper to sniper, pulse rifle to pulse rifle, etc...)
Not sure how the attack value works but I guess higher attack just means more dmg to pve enemies.
I doubt impact is the damage per shot. My rare sniper level 18 with 219 attack has half the impact as my level 16 rare pulse rifle with 215 attack. Also it would mean that rocket launchers don't do any damage since they don't have an impact stat.
