My understanding is that there isn't a difference, other than a couple perks with using the Wii version. Keep in mind I don't have the Gamecube component cables, so I'm definitely not authortative on the subject.
If you have the component cables for the Gamecube, games still won't use the progressive scan option (if the game includes it) unless you hold the B button while it's starting up. With the Wii, it automatically detects that it's a component cable and starts up the game with the better video options, again, if the game has that feature.
The one negative I can think of, and this doesn't deal with video quality itself, is that the Wii doesn't have the equivalent of the Gameboy player that the Gamecube does. So, you can't play Gameboy games, etc. using a component cable like the Gamecube can. Then again, I'm not sure if the Gameboy player even looks any better with component cables....
In my experience though, the Wii is a lot more finicky actually reading Gamecube games than a Gamecube itself. I recently played through Super Mario Sunshine and my Wii would occasionally (then later on very frequently) have issues with it. Popped it into my Gamecube and played through the whole thing no problem. Luckily I found another copy over the weekend and that plays perfect on the Wii
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