Hi There
In my CCNP (BCMSN) studies I learned that a backup port serves as a secondary "Designated" port. It will be in discarding state until the designated port fails, at which time it will transition to the forwarding state. A Backup prt receives more useful BPDU's from the same Switch/Bridge that it is on.
An Alternate port serves as a secondary "Root" port. It will also be in discarding state until the primary Root port fails, at which time it will be transitioned to the forwarding state. An Alternate port receives more useful BPDU's from another Switch/Bridge.
See this Cisco Document for some more details http://www.cisco.com/warp/public/473/146.html#states
HTH
Best Regards,
Michael