just remain working without any changes (with minor regressions
maybe)?
You shouldn't have any issues as long as you replace with the same version. If you're currently are using MySQL 5.1, you can replace with MariaDB 5.1. If you are using 5.5, use the corresponding one. MariaDB v5.2 and v5.3 and enhanced versions of 5.1.
Or will I have to replace/reconfigure client drivers (like use another
JDBC driver class and connection string)?
You shouldn't. MySQL and MariaDB both use the same protocol and overall syntax, but there could be a few exceptions in some cases. The URL quoted in another answer is a great resource: http://kb.askmonty.org/v/mariadb-versus-mysql
Regarding the drivers' licenses, there are differences if you plan to distribute your project commercially. In that case I recommend you to double check the license terms and if you have any concerns, use the MariaDB/SkySQL versions of the connectors as suggested in another answer. Check http://www.skysql.com/downloads/java-cc-connectors-mariadb for details.
Or will I have even to change application code?
The application code should be unchanged.