Shankha,
I just visited your site (2nd time today) and was automatically logged in as expected. I use Chrome on a Mac by default, so that's worked both times.
Everything I see looks setup correct, so I'm not sure what is wrong when you try. I think your problem is unrelated to the specific auto-login issue that this post is originally about though. Can you try:
* With a different Facebook user?
* Make sure that the User Map row in JFBConnect is correct for the user you're trying to login.
* Try disabling any caching you have enabled in Joomla. Clear your browser cache and try again.
Let us know if any of that helps.
Thanks,
Alex