If you think you'll want to use a cheap 50mm f/1.8 lens for low light photography or portraits, get the D80. The D60 can not use *any* of Nikon's Auto-Focus prime lenses. There's also a nice and relatively inexpensive back-catalog of AF tele-zoom lenses, dedicated macro lenses, etc. that will only work on the D80. So if you want to build a lens collection on the cheap using eBay, the D80 is your camera. The newer stuff (apart from prime lenses) will work on both cameras.
The D80's Auto-Focus system is also more advanced than the D60's in that it has 11 AF points vs. just 3. This makes the D80 a little bit better for action photography, etc.
The only other difference is that the D80 has an extra LCD screen and a few extra buttons. This allows you to change some settings a bit faster.
The image quality, resolution, high ISO settings, the size of the LCD screen on the back... all the rest is pretty much identical.
So if you're planning on keeping things simple with just one or two zoom lenses, you won't notice the difference. If you want a camera that you can 'grow into' spend a bit extra on the D80.
Here's a side-by-side of the specs:
http://www.dpreview.com/reviews/compare_post.asp?method=sidebyside&cameras=nikon_d60%2Cnikon_d80&show=all