In terms of hardware, GameCube CPU used copper interconnects, rather than the usual aluminium wiring to connect the chips millions of transistors, so benifited from newer chip manufacturing technology, that Sony did'nt have the luxury of, when producing the PS2.It was basically a customised Power PC 750 processor, just designed for console and thus was ideal as it used less power and produced less heat.Initally clocking in at 405 Mhz, Nintendo fine-tuned it to 485 Mhz for final production.
The GPU, like the cPU was designed to be easy to use, powerful, yet cheap to make.One of it's key strength's was the large amount of very fast on-chip memory for graphics processing, but even this was'nt enough for some developers,m see note later...
Whole hardware design was based around productivity over raw performance, basically you could get a game up and running quicker and using less tools on GC than say PS2 approach where Sony gave you a lot of raw power, but you had to allocate it where you saw fit yourself.
Lot of developers found the GC's Application Programming Interface (API) far easier to work with than say developing for PC/Xbox via Direct X, but you were limited to fixed function nature of the hardware, so if you wanted a custom effect on PS2, Sony's harware let you mould the raw power to create that effect, it just took time, with GC, you had to try and get fixed hardware to emulate an effect you wanted.
David Braben put the GC as being a lot faster than PS2, but was'nt overly happy with it's smaller main memory (Xbox had 64 Mb unified memory, PS2 32Mb Ram for graphics, GC only had 24 Mb for GPU, plus some fast on-chip memory)
Looks like GC VS PS2 was swings and roundabouts...PS2 had higher fillrate, faster at clipping polys to screen, but lacked the multi-texturing GC offered.Gamecube had better caching than Xbox, but if you wanted to do crazy-ass particle effects, or crazy maths as developers put it, raw power of PS2 was your best bet.
So in terms of which of the 3 was 'more powerful' really depended on what you were looking to achive in terms of visuals.