Replication can be a bit of a pain to set up compared to anything in the MySQL family since the tooling to manage it isn't part of the core project (there are tools out there, like repmgr from 2ndQuadrant).
Similar story with backups, bring your own tooling - again, 2ndQuadrant has a great solution with barman, there's also WAL-E if you want to backup to S3 along with many others.
Uber certainly presented a valid pain-point with the way indexes are handled compared to the MySQL family, any updates to an indexed field require an update to all indexes on the table (compared to MySQL which uses clustered indexes, as a result only specific indexes need to be updated). If you have a lot of indexes on your tables and update indexed values frequently you're going to see an increase in disk I/O.
Someone else can probably come up with a more exhaustive list, but the first two are things I've personally been frustrated with - even with the tooling provided by 2ndQuadrant I still have to admit other solutions (namely Microsoft SQL Server) have better stories around replication and backup management, though the edge is in user-friendly tooling and not so much underlying technology.
On the other hand, PostgreSQL has a lot of great quality of life features for database developers. pl/pgsql is really great to work with when you need to do heavy lifting in the database; composite types, arrays and domains are extremely useful for complex models and general manipulation; full-fat JSON support can be extremely useful for a variety of reasons, as can the XML features; PostGIS is king when it comes to spatial data; and a whole hell of a lot more.
PostgreSQL is hands-down my favorite database because it focuses on making my life, when wearing the database developer hat, a lot nicer. With the DBA hat on it complicates things some compared to other products, but the tooling out there is at least decent so it's not a huge deal.