If you do a seach on gameranking.com for the top rated games of all time, (any platform including PC, although they don't go back to any platforms before the Playstation, and limit your search to 50 games, you'll find 12 PS2 games, 2 X-Box games, and 1 Gamecube game. The X-Box games are Halo and NFL2K3. You already indicated that you don't like console shooters as much as PC (I'm with you there), and you can get the great Sega sports games on any of the consoles. If you care about a good quantity of quality games, the choice seems pretty easy to me. If you want to play some amazing games from the recent past (Final Fantasy 7,8,9, Metal Gear Solid, XenoGears, Vagrant Story, Chrono Cross, etc. a PS2 can play those too.
Fighting games, "Console games" i.e. Jak & Daxter, Super Mario Sunshine, Devil May Cry, etc. Driving games, and Sports games are the types of games I think consoles do better than PCs. PS2's Virtua Fighter 4 is probably the best fighter available; X-Box's Dead or Alive 3 doesn't show up in the top 100. While pretty, the gameplay is extremely shallow.
The X-Box does have better graphics, but honestly, there isn't a single game on it that I want to play that isn't on another console as well. If you are a sports game fanatic (EA or Sega, Football or Basketball), you don't really care about any other game genre, and you care about having the best graphics currently available then the X-box is for you. Other than that, I don't see why people bother.