Why do some items that increase attack speed work, and others don't?

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.