Arjan,
A couple possible causes:
* Clear your Joomla caches - Unlikely the issue, but worth a shot
* You already registered with that user and are simply logging in when you click the button - Probably not, but possible
* You have a Joomla account already with the same email address as the FB user that's registering. When you click the button, the "Automatically connect users by email" setting is enabled and kicking in.. automatically logging you in to the Joomla account.
Beyond that, not sure why the Normal registration wouldn't be used. So, try the above, and let me know if anything helps narrow things down. If not, let me know your site address so I can test, and we'll let you know what we find.
Thanks,
Alex