So far as I can tell, the Wii U can do almost everything a Wii can. The only exception I am aware of is the Wii u doesn't seem to have a game cube memory card reader. Is there a way to save GameCube games? And if so, can the info be transferred from existing cards?
The Wii U has no built-in GameCube support at all, similar to late-model Wiis - that's why there aren't any controller ports, or memory card ports. (The GameCube controller adapter only works with Super Smash Bros., at least for now.)
You'll need to keep a Wii or GameCube around to play your GameCube games.
While there is no official way to play GameCube discs on the Wii U, you can play GameCube backups using Nintendon-t (requires you to softmod the virtual Wii that the Wii U has).