Dealing with copied websites

Yes, you should copy the #__jfbconnect_xyz and #_onengraph_xyz tables from your previous site to your new on to ensure the best compatibility.

The #__jfbconnect_user_map table stores the links between a Joomla user and their Facebook credentials. If you don't copy that, then the user will have to recreate the link between their accounts. JFBConnect provides multiple ways to do that, like automatically linking users to their Joomla account if their email address is the same on their social network account.

One caveat to all of this is if you are changing your Facebook Application ID. Each Facebook Application will create a unique ID for each user. If you switch Applications, then the data in the user_map table won't be valid for your users anymore since the IDs will change. In that case, you don't really need to copy the tables since the data won't work for your users anyways.

I hope that helps answer your question,
Alex