Personally, the main advantage of Twitter is its ability to post a public text for all your friends/"followers" to see. It's identical to Facebook's status bar -- yet much more simple, and with the myriad of additional features like editing your profile, etc. I suspect the main reason for its exponential popularity lies in this interface model: specialize in one thing only, which in Twitter's case is simply allowing members to post a line about themselves or others (but mostly about themselves). It's the main reason Facebook trumped MySpace, as the latter offered too much customizations. Seems like people like the ability to customize, but only to a certain extent.
This news report from ESPN illustrates one of the great things about Twitter: allowing the direct communication between two individuals who otherwise would not have been able to. In the example, the owner of the NBA team Miami Heat is fined for purportedly violating the league's policies on discussing the current lockout. The $500,000 fine comes from a number of tweets the Heat owner, Micky Arison, responds directly to a number of questions from NBA fans. In short, his tweets mock the other owners and possibly reveal the growing rift in the opinion between all the 30 team owners. Humor nature of the example aside, it shows a billionaire talking directly to a....non-billionaire -- something that could probably not occur in Twitter's absence.
I might finally be signing up for Twitter, in the (very) near future. Why the change of heart? There are two reasons. The first is that Google recently announced the shutdown of its "Buzz" feature, which was quintessentially Twitter-on-Gmail. I don't "Buzz" a lot, but I think I will miss it. Second reason is the real possibility of obtaining a new smartphone -- with a data plan. This means I'd be able to tweet from anywhere, which is something I am not currently able to achieve.