It is a fact that the Wii is basically a refurbished Gamecube, this has been confirmed many times. Even most components are basically identical, with the major difference being that both the cpu and gpu of the Wii are clocked 50% higher than they were in the Gamecube. This is only logical since the production process hade been improved over the years.
Also the Wii has a total of 88mb compared to the Gamecube's 40mb, but again using the same architecture.
So again yes the Wii is more powerfull than the Gamecube, but it's mostly the same machine. Furthermore in some parts like the cpu even the old Xbox is said to be slightly more powerfull than a Wii.