I love Twitter. And I've written about why. I also struggle with Twitter... and I've written about that too.
But one thing remains consistent with my Twitter usage: it happens on third-parties like Twhirl and Alert Thingy as I rarely visit Twitter.com (unless on my Blackberry).
I've used Twhirl and Alert Thingy heavily over the last few weeks and have a completely unsatisfied feeling. They are each great as way of pushing and pulling data from Twitter... but really, that's about all they offer. Sure, there are quick links / buttons for various actions like direct messaging, searching, etc... but there really is no available functionality that isn't offered at Twitter.com (or FriendFeed.com for that matter).
Twhirl and Alert Thingy each have some much data and activity running through them that they really have no reason to be merely an API service. They should both be revealing interesting data sets and trends. For instance, I would find Thwirl infintely more useful / valuable if it surfaced my network's must 'interesting' information - perhaps that's base on my network's reading habits, click trends, etc.... whatever, it is, showcase some information or trending that can only be accessed via Twhirl. After all, that's really the only way to distinguish yourself among the competition (and there happens to be tons) and the only way to leverage your massive usage.
A complete side note - I'd LOVE some sort of mobile integration!