I've been thinking about how I'll use burrows in the next version, and I think I have two things I'll do immediately with this new feature. Both are things I tried to do early on in my DF career and ditched due to how annoying the inside/outside feature worked.
1) Proper Grand Entrance with defensive measures installed: When I first tried playing DF I tended to make grand entrace halls with guard stations and side passages alongside them with arrow slits and the like. However, because of the entrance dance and similar issues I abandoned these ideas. I do like them, though, and plan on playing with them in the new version. They won't be my primary defense measure, but they will allow me to put up a good fight should any hostiles get past the outer defenses. (I dislike traps for the most part, so I rely on soldiers and the occasional mechanical trap (flingifier and the like that takes effort to set up and use)
2) Hunting lodge: I've always wanted to have an area set aside for my hunters/woodcutters/etc outside of the main fort. The idea never worked as well as I'd have liked though, since the inhabitants usually ignored it except to sleep and random dwarves would decide to trek all the way out into the middle of the forest to use the dining room. Plus, it was very difficult to defend because I couldn't get them to hole up there, they'd either run around trying to do their jobs or run all the way back to the main fort. Neither option worked well.
Burrows should make all of this more manageable.
What about you? What things are you going to do with them?