I can't say I have a ton of experience with Mysql, but I am a huge fan of Postgres. I think especially for people who just want their database to work, having to worry about which of the storage engines you should be using for MySQL would be a headache. With Postgres, they focused on one storage engine and making it work as broadly as possible, and I think that is the right way to go about it. In addition, administration + configuration is really nice, the community and mailing lists are active and helpful, and the performance has been great for our usage. We also use it heavily for geospacial stuff (using PostGIS), and I think Postgres is ahead of MySQL for that.

I'm sure there are places online that do a much better job comparing them, but from my experiences with Postgres I would have no reason not to use it for any relational database needs.

I know MySQL has drunk the cool-aid with innodb, but 10 years ago their FAQ entry about why ACID was a bad idea convinced me I didn't ever want to rely on them for anything I might normally rely on a database for. I'm cool with intentionally trading off database features for other features that I value more (like availability), but I want it to be an intentional choice, and then it shouldn't still pretend to be a database for me - admit it's an unreliable data store and just use this unreliable fast thing for a cache, not as a database.