Reconcile users with logins using 'sp_change_users_login'
When you run sp_change_users_login, it automatically renames users as explained in Books Online:


  The name of the user will be automatically renamed to the login name
  if the following conditions are true. The user is a Windows user. The
  name is a Windows name (contains a backslash). No new name was
  specified. The current name differs from the login name.


To keep life easy, I'd keep the same usernames on both servers and then you should be fine with sp_change_users_login.