You will most likely have a SID issue. Unless your directory service admins ensured the SIDs were the same between the two domains (something that can be done during a migration), they aren't going to be because the SID is designed to be something which is unique.
That's why you have to script out the Windows logins for the old domains, do the domain replacement, and re-add them. When you do, SQL Server will have the correct SID, but this is still at the server level. You'll have to sync at the database level. This is where sp_change_users_login comes in. However, before you run it, you'll want to clean up the logins for the old domain. The database users will have their SIDs matching the old domain. Using sp_change_users_login once those old logins are cleaned up will allow you to associate those database users with the logins for the new domain.
If you are using different default databases other than master, you'll need to take that into account as the article says. You can script that out as well, or set it after you create the new logins, if there are only a handful you have to set.
K. Brian Kelley