Data backup and data archiving are systems designed to protect company processes and company data.  However, the reasons for performing data backup differ from the reasons for performing data archiving.  The main differences revolve around who will utilize the data and who drives the requirements for the processes.
Data Backup – Data backup focuses on preserving multiple point-in-time copies of active internal data for rapid recovery.  The goal is to make sure that valid data can be recovered promptly in the event of loss due to disaster, outage, criminal mischief, system corruption, human error, or other unforeseen interruption.  The business itself is the benefactor of backup data.
Data Archiving – Data archiving focuses on preserving a single provably correct copy of non-changing data that may be required for historical, legal, or external compliance reasons.  The point is to guarantee the preservation of information for so that it is available for internal, legal, or regulatory discovery processes.  While a company can benefit from archiving through primary storage offload and backup optimization, in many cases, external entities such as customers, courts, and regulators are the primary benefactors.
There is a whitepaper devoted to the topic.  You may find it here: Whitepaper: Backup vs Archiving