* Real-time breaking news
* Event planning and announcements
* Trial balloons for events, products, projects
* Humor
* Broadcasting the location of food trucks, or specials at a restaurant, or sudden openings at Alinea, &c
* Staying connected to broad but specific peer groups, like "everyone who does software security"
* Announcing and promoting content; specific example: getting a specific group of people to read a blog post
Facebook is good at very few of these things. Twitter is like a more efficient, more far-reaching version of the Facebook news feed; Facebook is a more intimate and conversational version of Twitter.
There has always been a weird rap on Twitter's lack of use cases, despite the fact that lots of people seem to find ways to use it effectively. A couple years ago, the rap was that the only reason to join Twitter was to follow celebrities. Nobody says that anymore; now it's "the only reason to join Twitter is if you're an extravert". Well, celebrities are 0.00001% of humanity, and extraverts are about half of it, so I'd say Twitter is making progress.