Hardware/GX
The Wii and GameCube's GPU is known as the GX. On the GameCube, it is also known as the "Flipper". On the Wii, it is sometimes (miscorrectly) called the Hollywood, which is actually the name of the chipset on the Wii, which contains the GX GPU.
The GX of the Wii is clocked higher than that of the GameCube, with nearly no other changes.
Wii-Linux sadly does not have any drivers to interface with the GX GPU on either the GameCube or the Wii.
A bit of information as to how this GPU works is available on it's WiiBrew page, as well as in YAGCD, specifically sections 5.11 and 8.