MySQL: Wait a sec, what do I use for this table, InnoDB or MyISAM? What are the tradeoffs again? Oh, right, do I want ACID transactions or fulltext search? Man, this tradeoff sucks. PostgreSQL: Wait a sec, which schema (namespace) should I put this table in? Man, it's nice to have namespaces within a DB.
InnoDB has supported fulltext indexes since 5.6 (GA release back in feb 2013). However, to be fair, fulltext indexes in InnoDB are not widely used and have a mixed reputation at best. Large MySQL-based companies tend to lean on external search services e.g. Solr.
I'm not familiar enough with pg's fulltext to compare. I would expect the functionality is superior to InnoDB's, but I am genuinely curious whether large pg-based companies rely on it solely vs using an external search index service.