There's a separate "Wii mode" for Wii and Virtual Console games. Supposedly you can't use the tablet or 360 controller in it; only the wii remote and wii accessories. So, expect less.
Wait, what? So I have to buy three sets of controllers to use the damn system?
I'm not sure what three sets you're referring to. But Nintendo expects you to already own a Wii Remote if you're buying a Wii U, so both versions of the Wii U only include the tablet. If you don't own a Wii Remote, you'll need one.
Where it gets tricky is with the Classic Controller situation. Wii U 360 Controller only works with certain Wii U games. Wii Classic Controller (Pro) only works with Wii/Virtual Console games in the "Wii mode."
It's still up in the air whether it supports the original Wii Remote for Wii U software. Nintendoland seems to require the Wii Remote Plus, but I don't think Tank Tank Tank uses motion controls at all.
In short, the nearly full experience: Wii U Tablet, Wii Remote Plus, Nunchuck, and Wii Classic Controller (Pro).