Similar to Facebook, Twitter is great for building relationships with library patrons and is not reliant on face to face interaction. It is also great for monitoring and responding to feedback and enquiries.
Although I personally do not use twitter, as Facebook is my preference, it is very easy to see how effectively libraries can connect with their patrons using Twitter.
With a 140 character word limit (nowhere near enough for what I like to write) libraries can shoot off quick snippets of information to their followers tweet feed. Also it is generally much more acceptable to post multiple tweets in a day, where on Facebook, people tend to get annoyed if you post more than a couple.
Twitter is great for communicating opening hours, special closures, and upcoming events.
Twitter can be used well for promoting new books, authors and spreading news about important library related events such as Cooking for Copyright.
Libraries can easily post photos and videos, provide links, and promote new items in their collection.