I was using one of the clients which were broken by the API change 2 years ago. Using it, I had a running chronological list of tweets, which made it useful for up-to-date information (travel news, road closures, for example).
That was impossible with the official client and the web interface, so I stopped using Twitter, since it had become almost useless.
Having checked recently, you STILL get a list of tweets in the order determined by Artificial Stupidity.