Naturally with any kind of outsourced function you're putting more dependency and trust on external services. So in this case, you're looking at handing over the backup and restore function to an external service provider. But with that you're also looking at dependencies on your Internet or your network service providers to make sure that your data can be transferred back and forth.
Once you go into a model like this for managed backup services, the backup plan can be somewhat complicated in terms of what do you do when you decide to move to another service provider or bring it back in house. How do you get your data back? Are you going to run into any termination clauses? Do you have end-of-contract fees that are going to be expensive?
Another drawback, that can be solved with good planning and due diligence, is making sure the server provider and their solution will scale with the growth of your organization.
Check out the entire Backup as a Service FAQ.