I am wondering what is the purpose of the "Game Masters" group in the livemap config?
Why can't I delete it if I don't need it? What's the difference between this and any other custom group that I add?

The default Game Masters group has a special filter that includes all characters who are permanent GMs on the server. These accounts have the "IsGM" column in the account table set to 1 and therefore can use the /gm command without password in the game.
That's the difference, and because of this function, the group can't be deleted.
If you don't need it, just disable tags and login.