Xbox kicks GC's ass in terms of exclusives. Pretty much all GC "exclusives" of note are either a. not exclusive (killer 7, resident evil 4, mgs: twin snakes is a remake of mgs1) or have better versions elsewhere (mario sunshine -> mario galaxy). The main exception is Metroid Prime, maybe Eternal Darkness, and a few others.
Xbox has Dead or Alive 2 Ultimate, Jet Set Radio Future, Otogi 1 and 2, Chronicles of Riddick, Phantom Dust, Panzer Dragoon Orta, Breakdown, Spikeout, Halo 1, Ninja Gaiden Black, and more, combined with the best versions of multi console titles. Most multi console titles during that generation were usually dog shit on the ps2. That's where the Xbox comes in.
I don't take anyone who says the Gamecube was a better system than the Xbox seriously on this matter. I know we all have subjective tastes, but in my experience anyone who says that gc > xbox is either deluded, a weeaboo, or both.