The Wii plays GameCube games and accepts the controllers and memory cards, the Wii U doesn't. I assume removing the ports and the mechanism to load small discs in the disc slot were the main reasons for dropping it.
The Wii was compatible with the GameCube largely by being basically an enhanced GameCube under the hood, and the Wii U is kind of a further step down that same road with its "vWii" backwards compatibility mode, so it's possible to use homebrew to play dumped GameCube games on a Wii U natively, only providing some light glue to hook things together. It's not officially supported though.
