<p>How do the full text search systems of PostgreSQL and MySQL compare? Is any clearly better than the oder? In which way are they different?</p>

Update:
Starting in MySQL 5.6, InnoDB
supports fulltext search
I'm not well versed in PostgreSQL unfortunately, but if you use the FULL TEXT search in MySQL you're immediately tied to MyISAM. If you want to use InnoDB (and if ACID compliance means anything to you, you should be using InnoDB) you're stuck using other solutions.
Two popular alternatives that are often rolled out are
Lucene
(an apache project with a Zend module if you're using PHP) and
Sphinx
.