The number of streaming channels is the same when you're connected to WiFi whether you're at home or out and about (it's currently possible to stream Sky Movies and Sky Sports only over mobile data if they're in your subscription).
It's not possible to stream TiVo recordings via TV Anywhere, but when you're at home and your TiVo box is connected to your home network you can use the TV Anywhere app as a remote control to select and start recordings to view on the TV that's connected to TiVo.
You can also use TV Anywhere to manage your TiVo recordings as long as your device has a data connection.