Some people have split accounts that you can choose to follow to receive different kind of status updates from them, so an example of this is a person with a 'oh look this interesting thing happened in my life' account, and a separate account for 'oh jeez I'm really angry and need to vent about this to a subset of people I know in relative privacy'. Because of the nature of twitter it feels easier to throw away or make lots of accounts than, say, facebok.
The real connection comes not when you follow celebrities, but when you're a creator (Or a budding creator), and you interact with other people who are creating things for support or advice; or you have an interest in what your friends are posting and doing, and like viewing their traveling posts and their reactions to things. If you have those sorts of relationships twitter helps facilitate them extremely well, and if you don't, or the people you interact with in that way don't use twitter, then it won't have that value for you.
I think warcher has a very good point about vocal people from other platforms (e.g journalists) giving a massive misconception over Twitters actual appeal and reach.
There is a huge difference between a platform where you shout in to the void and people have to choose to listen and a platform where you shout into the void and people can overhear you. One of those systems favours the popular, the other favours the average person.
Twitter gets a lot of love from journalists and people with a profile, who write the articles about twitter and control public perception. They have an interest in maintaining a personal brand and have enough visibility that Twitter works for them.
If you listened to those people, Twitter's biggest problem would be trolls writing mean (and often abusive) tweets about them.
Most twitterers would be lucky to be threatened with the rape and murder of their families, frankly.
Which is why Twitter is dying-- there are so, so, so many news curation and aggregation sources. Twitter isn't even all that good, frankly. Reddit does the job way better. (Still a lot of folks that want to fuck my mom though.)
As a social network, it's crap, has been crap and likely will remain crap. The time-based feed tends towards livelock from professional content producers. Their curation mostly suggests people I can follow who would generally prefer I do not engage with them beyond promotion of their brands and gentle applause.