The biggest difference—or at least the one that hits me biggest—is that Discourse by default shows you a flat view of the entire community. I had a hard time with this at first, but then I took a step back and thought about it, and it makes a hell of a lot of sense.
What’s the first thing I check when I visit a forum? New replies to topics I’m following. Discourse eliminates the step of first showing you the top-level category view, and immediately starts you off looking at new topics.
I’m not convinced this is the right way to go with a very high-volume site, but for medium- or low-volume forums, this seems ideal.