Backup: A collection of data stored on (usually removable) non-volatile storage media for purposes of recovery in case the original copy of data is lost or becomes inaccessible; also called a backup copy.
Archive: A collection of data objects, perhaps with associated metadata, in a storage system whose primary purpose is the long-term preservation and retention of that data.
In summary, backup and archive are two processes that solve very different problems. It is not uncommon to find customers using both in a complementary fashion. Backups are used as the primary method to protect corporate data and to enable large scale recoveries when needed. Archives, in contrast, enable cost effective retention and rapid access to important information for compliance or cost savings purposes.