The C-stick functions as both a camera controller and a quick input button, which is something I don't think I've seen in gaming. The PlayStation controller functions similarly, but not nearly as efficiently as a button, or as well in fighting games.
It doesn't respond anywhere near as quickly, and you have to force it in there, which thus leads to a significant increase in mistakes, especially if a game is high paced like Smash 64 and Melee (and not slower paced and about timing like Brawl). Also, the angles for it are really funky, and pushing in the wrong input is a lot easier.
If you doubt me, just try using a Classic Controller Pro, a WiiU Pro Controller, or a GameCube controller's control sticks and compare it to the GameCube controller's C-stick. I've done the comparison, and none of those controllers' control sticks comes close to the direct input for attacking and functioning as a quick/sleek smash bros. button as the C-stick, especially in context of how it is used in smash bros.