The D60 has twenty more "D"s than the D40.
Sorry, I'm feeling a little playful. I have no experience with these models.
LOL fantastic!
I'm a Canon guy, so I don't know much about Nikons. From what I gather though:
* 10 megapixel sensor vs. 6
* Lower base sensitivity (ISO 100 vs ISO 200)
* Flash sync 1/250 vs. 1/500 sec
* Auto ISO option includes ISO 200
* Faster continuous shooting (3 fps vs. 2.5 fps)
* EXPEED processing
* Image sensor cleaning system with Airflow control system
* Automatically rotating user interface (for using camera vertically orientated)
* Stop motion movie function
* Eye sensor
* Electronic 'rangefinder' for manual focus
* Active D-Lighting with dedicated button
* In-camera Raw processing
* New JPEG retouch effect
If money's an issue, I'd suggest the D40, as it's a very solid beginner's DSLR. If it's not, maybe take a look at the new Nikon d5000? It's a significant jump from the D40/D60, as both are about to be discontinued in the near future.
Hopefully some of the Nikon DISers can give some first hand experience.