One thing that PostgreSQL can do that mysql cannot (so far as I know) is lets you define your own triggers and even your own data types. In pretty much whatever language you choose.

PL/Ruby http://moulon.inra.fr/ruby/plruby.html for instance, could be used to define an insert trigger that would check the values being inserted into a table and perform arbitrary actions based on them; say, an alert if a user on a watchlist (another table in the db) changes certain values in their profile entry, for example.

It can lead to a fairly clean coding style where data consistency and manipulation code lives in the database.