Some time ago, I gave up on the notion that I could be an artist and have a completely private life. The thing with being an artist (musician, writer, painter, director, actor, etc...) is that you need fans. You need supporters. You need people to know about your artwork and support your artwork (both financially and intellectually). So, unless you have an agent working for you, you have to do this yourself.
I think the same applies to organizations, businesses, etc. How do you stand out in a crowd? One way is to use social media to build a fan base (or user community or support network or ... )
I twitter for 3 reasons: to get up to date info/links/buzz about technology through people I follow, to build a network of followers who I might not want to facebook (facebook is more personal to me although there is some overlap on fb) or those that might not want to facebook with me, and to post news/announcements relating to artwork or my writings (in other words, straight up PR).
There are of course, other reasons to use twitter. If you are confused about twitter, I suggest you take a look at this great post. It explains what twitter is, how to use it and why it is useful, especially as relates to organizations.