Should I use GDB or FDB extension?

Short answer: use FDB

Long answer:

On non-Windows platforms it does not matter much. GDB has historically been an extension for InterBase databases. Firebird is a fork of InterBase 6.0, so it inherited that. However, Windows systems since Windows XP have a System Restore feature, and GBD extension is included in extensions 'protected' by it:

http://msdn.microsoft.com/en-us/library/aa378870(VS.85).aspx

This means that database file will be backed up by Windows any time any write is done to it. Since Firebird writes information about each transaction in the file, this process will happen all the time. If you have a large database file, this can seriously affect performance.

This is the main reason. Beside that, Firebird is getting more and more different from InterBase and newer ODS versions (database file structure) and no longer compatible. So it is good to make it clear that database is no longer InterBase compatible file.

Firebird developers decided to use .fdb extension for new security database, so this is now a common extension to be used. Please note that there is nothing special about it, so you could use any extension and the database will work.