I prefer Apps which don't need an internet connection to use since the internet is not always available, but for gaming apps the internet is needed for online multiplayer, and high score charts, so if it is a game App I prefer an internet version, but with offline single player.
E-mail, Skype, Netflix, Hulu, Crackle, and YouTube App need the internet.
I think internet connected Apps are more cool since it save space since you don't have to have a lot of space to watch tons of movies, and you can play multi-player games, post and chat with people in realtime on the internet which is not possible with offline apps.
But, for simple apps like a picture viewer, games, and video player offline apps are good when the internet is off.