Bear in mind, this is when Nintendo did give it their all for graphics but the Gamecube was still weaker than Xbox.
Not really as Nintendo really did hit a jackpot then.
GC managed to pull more polygons per second by a great margin and being a texturing beast; something XBox really wasn't (and could only pass 4 textures per pass vs GC's 8).
Because in the end that was something developers really wanted/needed, it literally made it so that throughout the generation Xbox was stuck doing the "polygon trick"; which is rendering the scene twice just so it can manage to match the 8 textures per pass GC did for free.
It meant that they were halving the already lower attainable polycount; which is why Halo 1 was a 10 million polygon game and then Halo 2 actually had less polygons going on just so it could have bump mapping on every surface.
It also didn't have dedicated memory for framebuffer and texture cache and that made it so that it had to "fish" for it on the main memory bank, framebuffer alone would amount to 8 MB at all times, in the end it still had more memory to work with but it had much higher latency and bottlenecks going on; the whole multipliers balancing act really paid off on Gamecube in making it efficient just so it didn't have to wait or loose cycles waiting for stuff.
Of course both platforms had advantages and disadvantages, for instance Gamecube did EMBM basically for free (used on every surface in Mario Galaxy) and that would be preferable to use on it than DOT3 (which I reckon was an incomplete implementation too); with Xbox it was the contrary, the type of bump mapping that would be cheap for GC would be hefty for it, and vice versa, so one had to take that into account. Xbox also had some problems with stuff like transparencies; they were pricey on it.
Then again, Xbox was a home PC GPU, so it was actually designed to go over 480p (and it did); AA was also easier to pull since despite loaning RAM from the main bank, it had more of it. Feature set was also more modern and documented, but the GC was hardwired to match a lot of those features on the cheap anyway; for anyone that wanted to loose their time on it, that is.
I definitely wouldn't say Xbox was the most powerful; that was the notion back then (helped by Microsoft itself, as it has been said that they gave developers incentives just so their version of the game in multiplatform was better; which would make sense, otherwise why would they bother? they don't bother today and certainly didn't bother on GC a lot of the times despite there being scenarios where PS2 was 30 fps, GC being based on it was 60 fps with better water, and then you had the Xbox version which was the prettier but stuck on 30 frames per second again) and it certainly did have more expensive parts going on, yes, but Gamecube certainly was doing more with less to the point of saying that whatever was there on paper for Xbox didn't really apply to the real-world scenarios; or surpass those of Gamecube on a steady principle.