Gameplay seems identical to me. I don't think I've ever heard of others noticing anything either. GameCubes are super cheap and durable though.
The only main difference I can think of is their video ouput, which can be important, depending on the type of monitor you want to play on. If you have a CRT, it doesn't really matter, since you can just use the regular composite video (Red/White/Yellow cables) for either system. But if you want to play on a flat screen, you'll want to use a Wii, because the Wii can output 480p (progressive display). If you use 480i (interlaced) with either a Wii or GameCube with a flatscreen, you'll get some noticeable lag.
You can change the Wii's video settngs by going to Settings, Screen, and then TV Resolution. Then when you start Melee the game will ask if you want to start the game in progressive, and so you just hit yes. Under "Widescreen Settings" you can also change the aspect ratio to 16:9 if your tv/monitor is widescreen.
