I prefer GH for just playing guitar alone as well. This is mostly because of the strum bar first and foremost, the interface for guitar (like hammer ons), and to a lesser extent the challenge and songlist. If you only plan on playing guitar by yourself, I would recommend GH because of the reasons above as well as the price. But those are just my preferences.
There is no online in the Wii version. No leaderboards or anything.
I don't feel the need to try it on a different TV. I was referring to hooking up a the Wii to a different TV as well, not just the instruments. I was just wondering if anybody else was having the same problem.