Say, you see a chain of replies. You click Show More replies a few times, then you want to see replies to a particular reply. You click on it. Suddenly everything disappears except the original tweet and replies to this reply. There's no way to go back to the previous state. You click back and only see the top replies again. For example go to https://twitter.com/chromakode/status/731942777131425792 and try to follow all the replies. If you don't want to throw your PC out of the window after a dozen tweets, I admire your patience.
Also, say you're viewing an individual tweet (like the one I linked above). For some reason, search is unavailable from this view. Why? Also if you accidentally click anywhere outside of that tweet, the entire page reloads with the profile of the person who made that tweet. Happens to me all the time.
What if there's an image in this tweet? You click on it and get yet another modal window that shows this image slightly enlarged. We now have two levels of modal windows and yet we can't see the full image yet. You need to copy the image's url and add :large or :orig to the end of it to actually see the uncropped, full-size image.
I swear, a kindergartener could've come up with a better UI experience.