I've never used a Twitter client before, so when I saw a notice about Brizzly, I went to sign up for an invitation right away. I was surprised, of course, to get one back so quickly, but now that I've been using the service for a few days, I can definitely see the appeal.
(I've heard tons about TweetDeck and, to be fair, I do use Twitterrific on the phone, just for the sake of full disclosure.)
Brizzly offers some great functions, such as automatically refreshing, which is something that Twitter really should have automated. Additionally, it explains the trending topics in a way that really makes sense, especially sometimes when I look at things and think, "That person is not a celebrity...why are they being talked about?" The ability to create groups is a really great function as well, though I think, to be honest, I'll use it more to filter people out as opposed to truly 'grouping' people. Honestly, I couldn't care less about saving my searches, but the last feature that really makes sense is organizing the direct messages in the threaded message style.
You can sign up for a Brizzly account at the home page, but if you don't feel like waiting at all, hit me up and I'll send you an invite. It's a good look at what Twitter could (and should) be, and most of these features will probably eventually be adapted, but why should we have to wait?