Xbox was hugely more powerful than Gamecube and was capable of games that neither the ps2 and Gamecube could do for various technical reasons.

Most multiformat games were superior on xbox to ps2 and gamecube. Sometimes a night and day difference. The gflops performance of the xbox gpu was something like 20-25 where as the gamecube was 8 gflops. Not only that but it had a more advanced feature set and had 32bit colour output for transparencies etc. It trounces the wii on almost every level let alone its own generation.

Great looking games on xbox are far cry, conker, Halo series of course. Ambitious games not possible on ps2 or gamecube would be games like Half Life 2.

Some games were 720p on original xbox like soul caliber 2 which were 480i/p on ps2 and gamecube.

Don't get me wrong though I was a huge fan of gamecube but the xbox delivered a technical performance level well beyond ps2 and gamecube for many games. The build in hard drive hugely supplemented the memory and optical storage to allow fast streaming of data, something not possible on ps2 or gamecube. The xbox had 64MB of shared memory but the ps2 only had 32MB with 4MB for video and the gamecube had 24MB of main memory and 3MB of video. The gamecube also had 16MB of slow memory which was used to cache the optical drive so meant the optical drive speed of gamecube was blisteringly fast but that memory wasn't available as main memory in the same way.

A typical gamecube game vs the xbox version would be cut down or removed video in the gamecube version due to lack of storage, missing graphic data and assets in game because of limited memory and simplified 2 channel sound due to not having the true 5.1 sound output of the original xbox.

https://www.youtube.com/watch?v=UDtKLmiGwj0

https://www.youtube.com/watch?v=5suGMyqb-gY

The xbox was a hideous lump of a console but capable of some truly stunning games for its time but then it was really just a pc pretending to be a games console.

Halo, Halo 2, Star Wars: Knights of the Old Republic, Star Wars: Knights of the Old Republic II: Sith Lords, The Elder Scrolls III: Morrowind and superior version of the Splinter Cell Trilogy. And GTA of course.

Because of pc like power
Full thirdparty support
Fantastic first party games
HALO

Halo

Xbox had more good/great games:

Sure, many of them were also available for PS2 and/or PC, but outside of these game forums most people only buy one game system... for them the exclusive/multiplatform argument doesn't matter.

And for gamers who don't play PC, they got the prettiest console version of multiplatform games on Xbox.

GoldenEye saved the N64. It really did. If that game didn't come out in 1997, the system would've fallen apart during the basically 2-year wait between Mario 64 and Zelda: OoT.

What happened with the GameCube is they gave away the console FPS audience to Microsoft by not countering Halo and selling off Rare on top of that.

And Nintendo shot themselves in the foot repeatedly with the GameCube by making it look like a purple kid's toy, not having a proper Mario game at launch, cell-shading Zelda, etc. etc. etc.

GTA, Halo, Battlefield, Elderscrolls, Tom Clancy series, Unreal, Fable, Knights of the old republic, Doom, Half life etc

Xbox had an actual niche gaming wise, catering to a "mature" audience and bringing over PC experiences which no other consoles had like open world RPGs and competitive online FPS (essentially the future kings of the software market). Comparably gamecube's niche was simply having Nintendo games. It didn't have better platformers then the PS2, It didn't have better actions games, it didn't have better RPGs, it didn't have better adventure games, it didn't have better fps, it didn't have genres the PS2 didn't have or that the PS2 was lacking in... If theres one thing you could say it did have, it was more quality local multiplayer games (Smash, Mario Kart, mario party), but we've seen how those experiences diminished in importance as time went forward and its not like the PS2 lacked local multiplayer games

To repeat someone else, the Xbox was truly new 

Halo and the "Microsoft" brand name helped the Xbox out, alot in America.

Halo, Live, key 3rd party games such as GTA.

But also Nintendo also fucked up in many ways...

The unit itself looked like something designed by Fisher Price, the pad wasn't conventional and most of their 1st party offerings were quite weak.

Sunshine and Double Dash were a bit of a dissapointment to many and people didn't (and still don't) want cartoon Zelda. Just look at that Twilight Princess E3 reveal again for proof.

Also selling Rare was a massive mistake. Nintendo had a very decent and popular shooter in Perfect Dark, it wouldn't have stopped Halo in its tracks but it certainly would have helped.

Wasn't there also a black GameCube-version available from Day 1?

I like my Metroid-Edition:

And the standard Xbox wasn't much prettier than the purple GC:

Gladly I waited for the Crystal Edition, which ain't so ugly greeeeen: