Hendrik,
I couldn't find the Login with Facebook button on the page you posted. Can you point to the actual login page, please?
Also, when you say the normal Joomla login has problems, what are the problems? Do you get an error message, or some other behavior you can describe? JFBConnect doesn't alter the normal Joomla login flow, so if you're having issues with it, it likely isn't related to JFBConnect. I'd recommend disabling the JFBCSystem plugin to see if the problem goes away. If not, JFBConnect is not the cause.
Thanks,
Alex