Here are some of the differences that I'm aware of:
D60 adds:
Higher resolution (10 mp vs 6 mp)
In Camera RAW processing
Lower ISO capability (100 vs 200)
Slower minimum flash sync speed (1/200 for the D60 vs 1/500 for the D60)
Rotates the user interface when you rotate the camera
Overall, not very darn much. Don't forget the D40x, which fits in between the two. I think it's essentially identical to the D60 in IQ with a few interface tweaks.
Oh, the kit lens with the D60 is now a VR (stabilized) lens.
As has already been pointed out, it will not autofocus with older Nikon lenses. If you plan on just using the kit lens and other modern zoom lenses, this won't be an issue. Nikon is transitioning from using focus motors in the camera body to using them in the lenses, and with this series of cameras, they no longer have the focus motor in the body. The one area where this might bite you is if you want to use a prime (non-zooming) lens (because these are often really good in low light). Nikon has yet to replace their main line of prime lenses with ones that have focus motors in the lens. That means you'd have to either buy a non-Nikon prime lens or manually focus the lens.