Two part question for you all.

First what do you use and why?

Second (and the reason I placed this in the coding forum), is how different are they from a web developer standpoint. Is it easy enough to make your feed work with both formats or should I have multiple feeds on my site for 1.0/2.0/Atom?

Feeds are used to allow people to "visit" your site and see what content has been updated without having to enter your URL. The feeds get aggregated into a feed reader so that someone can be instantly notified of new content across hundreds of sites without visiting each.

ATOM is more detailed than RSS. Using either is easy. Just do a short search and you'll be OK. I always provide feeds in the latest version of RSS and ATOM, but not multiple versions of the same spec.

WordPress's default provides both RSS2.0 and ATOM. If you don't want to compare just go both!