Features I use via third parties is mostly analytics. When to schedule tweets? Whom to follow/unfollow? Which accounts are fake/spam? Who favs/retweets me a lot? I would like to see some overlay in the UI which gives me stats about other users.
If I would have something to sell, then we enter the world of conversion tracking. Twitter should support A/B testing. The interface for teams managing a single account seems weak (haven't used that much).
Imho the core service of Twitter is to connect brands with people. It still does that job better than Facebook.
The second you allow the feed to be presented outside their control they loose any ability to control how people deal with it. It would be easy for a developer to remove the advertising in the feeds.