I've used the official app, rowi, gleek and a bunch of others. I'm by no means a power user so I find myself using the Twitter timeline in the people hub most and then if I need added features I use Rowi.
I've tried a few twitter apps on my Lumia 900 and have found rowi to be the best: -
1) Push notifications (Tile + Toast)
2) I like the double tap to view the tweet or single tap to choose if you wanna reply or save etc
3) It's quick to load new tweets
I've tried Gleek (UI isn't as nice and its SLOW loading new tweets) and the official Twitter app (not as nice as the rowi app) and the People Hub would be my preferred "app" however no push notifications push me back to rowi
If the People Hub had more instant notifications of new tweets or facebook mentions then i'd use the People Hub exclusively
I use the official Twitter app just because it does almost everything I need it to do. It lacks "reply all" function, and sometimes it does not display the full list of the people I follow and that sucks, hope they fix it.
Everyone should expect third-party Twitter clients to stagnate following Twitter's recent policy changes. Carbon's still got the best design and execution of that design, every other client I've tried has something icky.
I used to mainly use Carbon. After hearing that there isn't going to be further development I switched to Gleek. I like the statistics and the List management. I couldn't view by list with Carbon. Can Rowi do Lists?
I tried Rowi free a while back. But that was before their big update. I should at least give it a try. I do like that they also have a Windows 8 app, but the W8 app seemed a little sparse on features the last time I checked.