I use both. I like the mute feature in tweetbot, and also how when you click a link it just 'slides over' to a new page in tweetbot. On the official app, clicking a link opens safari.
I check on the official app mostly for the '@ connect' tab. It shows what tweets of yours are retweeted and favorited. New followers also show up as an entry.
If tweetbot could replicate the the '@' tab, I'd delete the official twitter app.