The ash catcher is just like another bong that you connect to your existing bong. All it does is filter smoke/ash through water once before it goes into the bong, where it is filtered again. The perc is another water filter, but its built-in inside the bong, usually in the tube. So when you smoke from your bong, the ash will still collect in the bottom water, but the smoke will be filtered through water one more time(with one perc, twice with two percs, etc.) before you inhale it.

I have used both, and in my opinion, the perc is the way to go. The ass catcher looks cool, but its kind of a pain the ass, and you gotta be really careful not to break it. The best bong hits I have taken were from a triple perc 36 in. PHX. FUCKING BEAUTIFUL