It's clear than MySQL is more popular database engine than Firebird. Almost all of MySQL's advantages stop here and Firebird's advantages come out:
Firebird is really free and community driven project. MySQL is owned by Oracle. and there are no reasons for Oracle further invest in MySQL because Oracle has commercial database. One of MySQL founders Michael "Monty" Widenius left Oracle by acquisition of MySQL because of that. Read his message: Help saving MySQL.
Firebird is more reliable. There are no index or table corruption problems so often with MySQL.
Firebird's database is just one file with .fdb file's extension (the same as in commercial Oracle's databases). So it's easy to move or backup database by zipping or moving the file. MySQL database in distributed in many names. It's hard to backup.
Firebird since the beginning has triggers and stored procedures while these construction were added to MySQL only recently.
Firebird has very small installation package. It supports embedded mode. Therefore installation of server is not required when working with local files.