The major benefits are giving people a way to offload the complexities associated with backups especially the nature of managing backup solutions from an application perspective, managing tapes, trying to contend with offsite vaulting of tapes. This is coupled with the need to lower capital and operational costs.
And then ultimately, the biggest driver is lower risk, especially for small to midsized businesses, mobile users or remote offices. Backup is often there, but in terms of effectiveness or a company's ability to restore data effectively, a lot of companies are just getting by and there is lot of concern about potential data loss.
Check out the entire Backup as a Service FAQ.