The new UI is just about the smallest change that Twitter possibly could have made. Now, tweets that are part of a conversation will be connected with a blue line. That's really about it. Only three replies will be shown on your home screen, but the blue line will tell you how many other messages there are in the conversation, and you can expand it if you'd like.
The updated Twitter app has already launched in both the Apple App Store and Google Play Store if you'd like to check it out. Of course, the web version of Twitter has also been updated.