There is just a little difference.
The "different server" is a different physical machine (which makes sense when using mirroring for high availabillity)
The "different server instance" is another installation of the SQL Server software which can be located on the same physical box (not helpfull for mirroring imho).
Eg. the default instance will be on port 1433 with a name of MSSQLSERVER. You would connect via management studio just with the DNS name of the server.
Every next instance will needs to have a different port to use and will also get a different instance name, eg. BACKUPSERVER. You will connect with management studio to this with DNS name of the server plus instance name, eg. SERVERNAME\BACKUPSERVER.