One illuminating example was a calculator application which had numerous errors and crashes which stemmed from intricacies of the UI implementation. By designing it as a statechart transitions are made visible and relationships are clear. The book provides three more examples.
In addition to the posted blog post, there are two more concepts in statecharts that I think are important: history and concurrency. They are not easy to explain in this comment, but the gist is that “history” allows for a simpler expression of returning to a previous state and “concurrency” formalizes the behavior of having concurrent states.
I’d recommend anyone to take a look at the book and I’d especially recommended to anyone who is a little skeptical of the idea of statecharts. I think the biggest argument against statecharts is that it requires you to specify your actual design, instead of just hacking on the code.