I've not used that many puzzles that have them, the only downside I can think of really is that it seems that they do tend to make a puzzle less finger trickable, and if you get a pop there's a chance that the ball bearings will fly out. But I do quite like the feel of clicking mechanisms, especially on the pyraminx for some reason.
I'm not sure why clicking mechanisms are used with some puzzles, is it just because the creator wants to go for a different feel? Or is it because clicking mechanisms make certain puzzles better in some way?
I sometimes wonder how the 3-layer Ayi's Cheese would feel without a clicking mechanism, I'm not sure if I would prefer it like that or not.
Clicking mechanisms are fantastic since they mean you never get misalignments. I am guessing they are not good for corner cutting though.