Srdjan,
Thanks for the credentials to the admin area of your site. We spent a lot of time trying to narrow down why the user wasn't being logged in during the registration process, but never got a full grip of it. Basically, what we were seeing was:
1) User clicks login button
2) JFBConnect creates the user correctly
3) JFBConnect logs that user - this works, you can see the user is logged into by going to the User Manager in the admin area
4) JFBConnect redirects the user to the page they were on
5) During this redirect, when the new page loads, something is clearing the user's session, which effectively logs them out
We can't determine what's causing step #5 above, and tried quite hard to figure it out, without disabling too much on your site to make things bad for any visitors that may have been on.
As a workaround, we enabled the "Auto-login Facebook Users" option, which will automatically log them in after the redirect. It's a strange thing to have to enable to fix this, but it does work. It will, however, mean that any user's that have connected to your site through Facebook will be automatically logged in. This may or may not be a good thing.
Hope this helps get you started though, and we'll continue looking at our code to see if there's anything we can come up with. If you know of any extensions on your site that may check the login state of the user and could log that user out for some reason, please let us know.
Thanks, and good luck,
Alex