Resident Evil 4 is better on the GameCube, the graphics are better and the controller suits it better as well. There is far less detail in the PS2 version, environments look bland, and blurry.
Metal Gear Solid 2 is better on the Xbox. It's a lot easier controlling Snake with the Xbox controller than fighting with the PS2's horribly placed, inaccurate control sticks. Any time you play a game on the PS2, the enemies are only half the battle - the other half is fighting with the controller. The loading times are also a little faster I believe, and there's a slight increase in visual quality.
Any Sonic game is better on the GameCube. Even the Dreamcast ports, the Sonic Adventures. They have added features, there's a slight visual increase, and the GameCube controller is just a lot more comfortable than the Dreamcast controller. It's just nostalgia talking when anyone says the Dreamcast versions are better; there's simply nothing the Dreamcast versions have that the GameCube versions don't, in fact, they lack things.
Resident Evil 2 is better on the GameCube.
Psychonauts is probably the best on the PC with an Xbox 360 controller, but if you don't have one that can run it, the Xbox version is the best.
The Prince of Persia series and Beyond Good & Evil are best on the GameCube.
The Grand Theft Auto series is the best on the PC - better graphics, and modifications! But if you don't have a good gaming PC, then the best console version are the Xbox versions - faster loading times, GTA3 and Vice City have upped graphics, and all of them have custom soundtracks. Then there's the fact that the controller just suits them better than the DualShock.
If you don't want online, the Tony Hawk series is easily the best on the GameCube. The controller suits it very well. If you cannot live without online, you can fight with the DualShock I suppose.
ALL Tom Clancy games are better on the Xbox or PC than any of the other systems. Period.
Shenmue 2 is better on the Xbox. You'd seriously have to be on something to think some import Dreamcast version is better. The Xbox version also comes with a second disc. Xbox version also comes with Shenmue The Movie.
Any first person shooter is going to be the best on the Xbox because the controller better suits it.
So the formula to follow is, when it comes to multiplatform games, they're almost always better on the Xbox, with a few exceptions in which they're better on the GameCube. I don't think I know of any multiplatform games that are better on the PS2.