Your live server has a bad php.ini setting for session_save_path. It is either a non-existent directory or it doesn't have adequate privileges, so session variables can't be set and so login won't work.
I created a set of test pages to demonstrate the problem:
http://www.nationwidejustsayyes.com/admin/setSession.php
You can forward that to your hosting provider and they should be able to fix it for you.