Attack speed increases animation time, thus gives you the ability to have more primary animations go off. (Primary animations are always forced to be displayed, secondary animations are only displayed if you are idle, for example the barbarians war cries)
In the case of channeling (a primary animation, that would be interrupted if you cast another primary animation, because only one can be displayed at a time) your damage ticks will occur more frequentlyb tu your resource wil drain faster as well, making very high attackspeed on channeled abilities rather useless. It'd be better to have a slow, hard hitting weapon, since most challens have pretty fast damage ticks already.
Summons benefit from attackspeed in two ways.
First your summoning animation is faster, meaning you can drop that Gargantuan in the enemies face quicker, to faster go and collect loot again.
Second, most pets scale proportional to your attack speed, meaning if your attack speed in the Stats-Page is higher, your pets will also hit faster.
(Also the Thasker and Theo's Gloves multiply with your attacckspeed, making your pets go berserk)
For Fire/Forget spells, you oughta read up on every spell you want to know about seperatly, because they are all individual. (Frozen orb will be casted faster but the spikes it shoots stay the same rate - Hydra will be casted faster but also attacks faster, etc.)
Finally damage over time spells only benefit from the cast animation (wich might be useful for Jade Harvester builds to get those dots in faster before you blow them up) but do not increase their damage if you run higher attackspeed. (Critchance and Critical hit Damage do increase the damage of dots)
I know this is a very old question, but seeing that the only answer was completely wrong in most points (dunno what's with that) i figured i'd go and set it right.
Hope I was helpful, guy from 3 years ago.
Tophatcrusader
The tooltip of the weapon already shows you the corrected dps accounting for the stats (any elemental damage, % based damage, faster attack speed) on the weapon (NB. str/dex/int - for the respective class - is excluded and only shows in the dps value of your character sheet)
Best Answer
This is the result of a current bug in the way Diablo III Handles Attack Speed enchants, as of version 1.0.1.9558. Essentially, there are two different enchants that increase attack speed, which work slightly differently.
The first kind, which displays on tooltips as Increases Attack Speed by x, is supposed to only show up on weapons. When this Affix is on a weapon, it increases the attack speed of that weapon, but it does not increase the speed of your other weapon if dual wielding. Most importantly, if this verbiage is found an item other than a weapon, it does nothing at all.
The second kind displays it's tooltip as Attack Speed Increased by x. This affix applies it's modifier globally, and will always work, no matter what item it's on. If this affix is found on a weapon, it will even apply to your other weapon while dual wielding.
This is slated to be fixed in patch 1.0.3.