There were very cool things I couldn't find a way to do in MySQL which eventually made the switch an amazing change. It was trivial to set up aggregated time series tables, and very easy to roll that data up in order to speed up queries with a coarser granularity. At the time I recall window functions making this really clean and easy. It was possible in MySQL (and maybe trivial today, too) but it felt like a massive kludge every step of the way.
This is actually what lead us to consulting someone who could implement the best solutions to these issues that they knew. They encouraged us to adopt Postgres before doing any work, we didn't listen, then as they wrapped up the project they once again advised we leave MySQL behind. He then sent some examples of how we might make the migration and why it would be better. Months later we had made the migration entirely and were very, very glad.
That guy was an AWESOME DBA, and he changed the way I value someone who is good with data and databases. He transformed our team's ability to grow our product, dramatically improved our ability to deliver, and instilled a ton of knowledge in a short time. I can only dream of imparting that to other teams!
Overall I'd say that Postgres provided tools that made working with data easier. I'm not sure if there are any specific features besides a cleaner, simpler,