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.