I think the main advantage of the wii is memory bandwidth, its huge compared to the original xbox I think. If I remember rightly the xbox unified memory is something like 6.4GB/sec but that is shared between gpu, cpu and any other processors and was high latency. I think there is a sound and i/o processor as well. The gamecube was 10.4gb/sec for its video memory and 2.6gb/sec for its main memory but the main memory had a very low latency. The wii upgrades everything by 50% over gamecube so video memory is upto 15.6gb/sec and main bandwidth is upto 3.9gb/sec all with very low latency and a few other features that increase bandwidth beyond that. The point is the wii is easily superior in memory bandwidth overall.
I know the wii loses the 16meg of slow memory the gamecube has which was used to buffer data coming from the optical drive. The gamecube only had a small disc with a single layer so 16meg of slow memory was sufficient to massively reduce load times. They had to replace this with the wii as it has a much larger disc with 2 layers so it was upgraded to 64meg of ddr memory. This isn't normal memory used by the main program but buffering/cache type memory to make access times from the optical drive much faster and also buffer sound data.
So the wii has 24meg of main memory, 3meg of video and a fully buffered optical drive.
The xbox has 64meg of main memory of which I think it can dedicate upto 12meg to video and has two streams storage data, one stream is from its optical drive with a small amount of buffering, the other stream is from its hard drive where each game can cache about 750meg of data from the hard drive.
The xbox has true 5.1 sound, a much more powerful gpu but probably a slightly inferior cpu to wii (marginal) and inferior memory bandwidth.
By looking at the wii specification you would expect the wii to offer more simplistic games to xbox with inferior graphics but be less prone to frame rate issues and I think that is actually the reality when you compare the games. This is also validated by the fact that xbox games are much larger than wii games. Most wii games are on single layer dvds, all xbox games are on dual layer dvds. I'm not saying all xbox games require dual layer though its just the way they are mastered. However xbox games are generally larger than wii games. Many wii games are no more than a few hundred meg compared to xbox games probably in the area of 2-4gig.
The conduit which is probably one of the most impressive wii games at times has quite a simplistic engine compared to Halo for example. Much smaller play area, reduced textures and a complete lack of ambition compared to to the xbox game. Halo was a early xbox game where as the conduit is a much later wii game. Shenmue on the dreamcast is a more impressive engine than the wii's conduit for textures but the dreamcast is limited by only have 16meg of main memory and a slower cdrom drive which means frequent loading pauses. However the dreamcast has 8meg of dedicated video memory compared to only 3meg on the wii. The dreamcast can't match the conduits excellent lighting effects though.
Here is an image from the conduit. The distant skyline is a obvious bitmap image which is pretty dated. There is obvious aliasing issues with the monument in the distance and on the ground.
This is the dreamcast image from shenmue
It looks much more natural. Diagonal lines like the fishing rod and metal framework are anti-aliased.
Here's a reminder of some of the incredible graphics of Halo;
A highly impressive scene which admittedly the open areas in halo can have a few frame rate issues but it doesn't effect gameplay at all. Its never too bad.