You should not use the CB Facebook connection option if you're using JFBConnect. There should only be one extension managing social network authentication on your site. Using two will only cause conflicts, though I don't think that's the issue you're having here.
I'm not sure what data isn't being imported properly, but the way to test and investigate further is to:
* Mark *all* profile fields in CB as *not* required
* In the JFBConnect -> Profiles area, remove the mappings of all fields from Facebook (or other social networks that you want to test) into CB
With that, JFBConnect should be creating the most simple user in CB with just the most basic information.
If that works, start adding new fields to the JFBConnect Profiles mapping area, one at a time, and see if there's any specific field that causes issues.
Then, finally, if that works, start setting whichever fields in CB to required that need to be that way.
With that, we'll be able to eliminate a specific field or determine if the whole creation process at it's most basic level is what's going wrong.
Finally, at the top, you mention you're using Joomla 3.03+CB 2.03+JFBC 6 . Can you let me know if you're really on Joomla 3.0.3? I'm assuming/hoping that was a typo, but please let me know the specific J! version you're using.
Thanks,
Alex