Kristoff,
A few things could be causing the problem. First off, did you run Autotune in JFBConnect for your Facebook App, or did you configure itself? I'm seeing the same problem you mention. If I go to the non-www version of your site, there is a Javascript message that your app may be configured incorrectly and that Facebook login won't work on the non-www version of your page. Autotune usually takes care of your settings so that things should work on the www and non-www versions of your page.
So, if you haven't already, run Autotune, go to the FB App section and hit "Refresh". Make sure all the recommendations are set or make modifications as necessary.
After that, try again. Possibly, also try with a different template, though I don't think that's the cause.
I'm also not sure what you mean when you say "The weird thing is that when I try to log in with the facebook link, it works. " What "log in with facebook link" works? I only see one, and it didn't work for me.
Finally, on an unrelated note, in the Javascript is the following message from Twitter:
TWITTER WIDGET: This widget is being deprecated, and will cease functioning soon. <Invalid consumer key/secret in configuration> widget.js:12
TWITTER WIDGET: You can obtain a new, upgraded widget from <Invalid consumer key/secret in configuration> widget.js:12
It sounds like you may need to start looking for a new Twitter stream widget soon.
Keep me posted on the above. Not sure what's wrong, but we'll help you get to the bottom of things.
Thanks,
Alex