PostgreSQL vs. MySQL: spatial feature comparison
I can't speak to advantages/disadvantages vis-a-vis MySQL, but the PostGIS code is pretty widely regarded as one of the best (in terms of speed/functionality) and most mature (in terms of testing/real-world exposure) available.  

By way of example, there was a talk at PGEast 2010 by some folks from the FAA on their converting their airport database (used by AeroNav and others to compile charts) to Postgres/PostGIS from Oracle.
The avationDB site is also built on top of Postgres (8.0).

If GIS-related queries are at the heart of what you're doing my suggestion would be to go with Postgres.  It can certainly handle everything else you would normally do in a relational database as well.



In terms of making the switch from MySQL, the documentation behind Postgres is first-rate, and there's also a section of the Oostgres Wiki about switching from MySQL to Postgres.
The initial learning curve may be a bit steep and you may need to tweak your database and any stored procedures (if you've written them for MySQL already), but it is not an insurmountable task.

You should be able to pick up enough to make the switch in a couple of weeks time, and if you set up a development database you can probably be well versed in routine tasks within a month, and confident you know where to look in the manual for the not-so-routine ones.