I think a lot of people completely miss the point of twitter - I find it a really fun and useful network, and one that I enjoy using much more than facebook.
Sure, it takes a bit more thought to post something meaningful and share-worthy on twitter, but I like that challenge. And because it's completely open, and not tied to confirmed friendships or links between accounts, it has the potential to send something viral.
So for your band, you could give away tickets to one of your gigs, and post the link on twitter. Chances are, it will reach people in your area who don't know about you, some of whom may come to your gig.
Plus, it's potential for anonymity (if that's your thing) means you get a lot more honesty on twitter.
Sure, you have to choose who to follow really carefully, but if you choose well, it's a really rewarding place to hang out.
For me, it's divided like this: use twitter to build an online 'brand' - over time people get a sense of your personality, and it's very informal. Anyone can talk to anyone, and it's a great place to expand your network.
Facebook is better for keeping in touch with the people you know in real life, and if you can stomach the endlessly-banal status updates of some of your friends, it has potential. Personally, I only really use it for the live chat function. Other than that, I prefer email.
If you fancy following my (usually quite ridiculous) tweets, hit me up @notbanksy
