The only difference is that you can play with friends/family member/strangers in your tour. You still get the points/stars for completing the songs etc. You reap the rewards if you pass, but you also risk losing fans if you fail. And it's a bit risky going online with your tour if you care for it because people may drop and if you're not near the end of the song that's a definite fail.

As I said earlier, it's the same thing as local but you're online.

Adding on to Calicoo's answer, the only difference with online tour is that you can have other players join your band, or you can join another player's band.

If you are playing alone or with other people in the room, you are NOT missing out on anything by picking one mode over another, as they both will serve as exactly the same thing. Don't be afraid to "commit" to one mode.

The one real difference is that local tour is as it implies for people that are there but if your alone you can always go onto online tour there are two different modes so that you can be able to know that there is a distinction