Personally, I don't feel as much depth when playing online games as I feel playing offline games. I can't think of playing a single player RPG or adventure game. It just won't have the same depth. I stand to be corrected though but I am yet to see an online game that has depth as much as its offline contemporary.
In all honesty, I prefer offline games more than the online games, but I enjoy both depending on my mood. One major thing that makes online game interesting to me is the competition other than that, I prefer offline games.