The Wii did have a few distinctions from the GameCube. However they abstracted them in an interesting way

The Wii itself was still a G3 PPC chip, just add a few hundred Mhz

It also had more RAM (which allowed the "Twiizers" attack over a decade ago)

All the Wii-centric functionality (USB, Bluetooth, Wi-Fi, SD card, etc) was abstracted away inside an ARM co-processor inside the GPU. Named "Starlet" by the homebrew community

Wii software was then written that would simply poke at the API's it exposed to add that functionality