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).