Just to expand on this, A-Click gives you an Attack Move command. That is, you attack the target clicked or attack the closest target to your champion in the direction you clicked. It's the latter that's important. If you miss right-clicking a minion or champion or whatever, you'll move next to them; if you miss A-Click a target, you'll just attack nearest target.
Basically, it's to keep accidental movements from happening.
The biggest thing you'd notice is Anivia's Q and Diana's Q as firing faster due to not having to hit Q, aim, click. You get to aim, Q. That's it. It removes a fraction of a second of delay. Like ... that's really it. It removes that extra click at the end but, sometimes, it's the most important thing in the world.
If you have him, the most obvious case of how great this is is with Ryze. He tends to mash all of the buttons (QRWEQEQWQEQ etc. etc.) so having to do Q-Click, R, W-Click, E-Click, Q-Click, E-Click, etc. 10+ times in an engage will eventually cost you.
It's not really trying to pigeonhole her so much as it was kind of absurd she got so much free Armor and MR out of Cougar form. Granted, they maybe could've just reduced it instead of lopping it all off but eh.
I can try it at some point. Nashor's Liandry's BotRK Kayle!