SMTP Error! Could not authenticate

That message will occur when sending emails fails. It is usually if the server is not set up correctly to send emails. JFBConnect attempts to send the email on new user registration (like when you just logged in with your social credentials). If you register via the standard Joomla registration form, I'd expect you to receive the same errors when you register.

You can
1) Check your Global Configuration settings for email configuration.
2) Check your email notification settings a) in Joomla backend > Users > Manage > Options button with 'New User Account Activation' and 'Send Mail to Administrators' and b) JFBConnect backend > Configuration > General > 'Send New User Email' setting.
3) Attempt a registration with a normal Joomla user account to see if the same behavior exists.

Try those and let us know if you can't narrow down the problem.
-Melissa