Restoring Differential Backups
In order to restore a differential backup, you need a database that is in "restoring" state. Which is to say that recovery can't have run yet. This is accomplished by adding the "norecovery" clause to your full backup restore. So, it goes a little something like this:

DECLARE @file VARCHAR(255) = '\\RemoteServer\Dir\\DBName_' + (select DATENAME(dw, DATEADD(day, DATEDIFF(day, 0, getDate()) - 1, 0))) + '.bak'
RESTORE DATABASE [DBName]
FROM DISK = @file
WITH REPLACE, NORECOVERY

--now you restore your differential backup