You might think of the Connect tab as your Twitter inbox. It used to be widely considered proper to respond to all interactions; that is, thank everyone for their RT's; for their favorites, when that came along; and for their mentions, e.g. #FollowFriday love.
As interactions became automated by Twitter, with the Retweet button, for example, some of the urge to respond fell away. It's still a nice gesture, though!