You CANNOT apply a differential without applying a FULL backup.

If I understand you correctly, you made a full backup and restored it to a 'backup' server. Now time has gone by and you want to update the information on the backup server; so you made a differential backup and want to apply it to the 'backup' server. Can't do it.

That's why log shipping and replication were created.

You can't apply a differential or transaction log backup to a database that is already operational. That's why when you are restoring several backups (FULL, Differential, and Logs) you have to use the WITH NORECOVERY option. Once the database recovers, that's it you can't restore any more backups to it.

-SQLBill