What's the difference between playing a GameCube game on a GameCube and a Wii?

The really aren't many differences at all. It's hardware level emulation, essentially a Wii has all the hardware that GameCube does. The only difference you'll notice when playing GameCube games is that with a Wii you'll have to go through the Wii's menu to start the game, where a real GameCube will boot right into the game.

Other than that the differences are in the GameCube specific hardware you can connect to the Wii. GameCube controllers and memory cards will work, along any other device that connects to the same ports, like the GameCube microphone or the Donkey Kong bongos. Other hardware that connects to other ports won't work because these ports don't exist on the Wii. Things like the broadband adapter that Michael Madsen mentioned, the modem adapter and the Game Boy Player won't work.

There's one big advantage that the Wii has over the GameCube in terms of hardware that can or can't be used with the console and that's component video cables. A real GameCube requires an expensive cable that was hard to find even when Nintendo was still making GameCubes. Component cables for the Wii however are cheap and easy to find. Component video cables provide much better video quality than composite video cables and allow playing games in 480p progressive scan, at least for the few GameCube games that support it.

Otherwise a GameCube game will look and play on a Wii exactly like it does on a GameCube. There's no quality improvements, the exact same performance, loading times and resolutions.

A Wii running in GameCube mode isn't emulating a GameCube, it is a GameCube - it reconfigures its hardware to run at the same clock speeds, etc.

There are a tiny number of differences:

• Some unlicensed software (e.g. FreeLoader) may not work, since the Wii has stricter checks against these (especially after Wii update 3.3).
• There's no way to hook up an expansion like the Broadband adapter, and the Wii doesn't emulate it (only useful for playing Phantasy Star Online on a private server, or exploiting the same game to run e.g. homebrew).

That's it, though.

If all you care about is playing games from your own region from your discs, a GameCube-compatible Wii is every bit as good as a GameCube. If you want more than that (games from outside your region, homebrew, backups), then you'd probably want a real GameCube (and the necessary extras).