I mostly use my Wii U, but I still have a Wii plugged in near it.
In all honesty I find the Wii U much more convenient to use. Booting into the vWii doesn't take too long (and you can force the Wii U to boot up into that rather than going through the Wii U's menu), native HDMI which is a nice plus, and every Wii game is guaranteed to work because of how Nintendo does backwards compatibility! In addition, Nintendont works great for playing my backed up GameCube games. For playing most games, it's probably the best thing to use, especially since the Wii U has some fantastic titles that I can go back to whenever I want without having to power down another console and switching my video output.
However, for creating backups of GameCube discs I use my Wii. CleanRip simply doesn't work with GameCube discs on the Wii U which is a bit of a shame. The Wii also has the slight advantage of having GameCube controller support. With this, you can play any game that has native GameCube support (Including Brawl and all of it's wonderful modifications, Resident Evil 4 Wii Edition, and Sonic Colors). Currently, nobody has been able to get the GameCube controller adapter for the Wii U to work with the vWii side of the console, meaning you have to use the a Classic Controller or the Wii Nun-chuck combo. I don't particularly like these two, but I hear the Pro Classic Controller had a much better design and HORI made an okay looking GameCube controller wannabe that acts like a Classic Controller.
If somebody could get the Official Wii U controller adapter working, I'd have no real reason to go back to my Wii other than to dump GameCube discs to play on Nintendont, but I've already dumped most of the my games anyway.