Why would it be better than MySQL is you're already using MySQL for years and built knowledge with it and have no issues? Why would I change, why Postgres would be better?
The JSON/JSONB columns and the JSON navigator options are amazing. MySQL isn't even close yet, they have a JSON column that's basically opaque. Other column types like the INET, CIDR and MACADDR are often invaluable.
You can also index ARRAY columns, meaning if you need a "tags" feature it's ridiculously easy to implement, no need for additional tables.
There's also PostGIS, which has no MySQL analog, and makes doing actual real-world GIS work possible.
Having tried, and failed to use MySQL Spatial Extension (which some might argue is PostGIS' analog) and failing miserably at some pretty basic things, I agree totally. I'd take PostGIS over ESRI's SDE and Oracle Spatial as well.
One "quality of life" thing - you can wrap changes to the schema in a transaction, so that a given set of changes is atomic. (Elixir's Ecto and Ruby's ActiveRecord both have migrations that do this by default.) As far as I know, that doesn't work in MySQL.
Feature for feature, PG blows MySQL out of the water. PG gives you way more tools to use in your queries, indexes etc. to solve app problems.
The real knowledge gap would come into things like all the various tactics you need to learn to get MySQL to produce good plans for your queries. PG has a much smarter optimizer which is harder to control - if it does the wrong thing, it's harder to encourage it to do the right thing, and philosophically they don't support query hints. OTOH, worst case in PG is usually much better than worst case on MySQL.
