Can we see the login code? Otherwise, there really isn't enough information here.
tail -f /var/log/httpd/error-log
Try to login and check the error log for errors
Check and make sure sessions are configured properly in PHP
Check the PHP version between the servers; it might be different enough to cause problems.
Make sure the MySQL DB exists on both servers and is configured with the correct credentials and is running.
Could go on and on. Provide more information and maybe we can help more specifically.