There's a lot of confusion out there regarding cloud for backup because you can use cloud storage to implement backup. To some people, their application for cloud is backup, whether it be regular backup, restore, business continuity, disaster recovery, or maybe a file sharing service like Amazon Zocalo, Dropbox or Box.
There can be differences architecturally, product-wise, in the deployment and services, and how they can be accessed. Think of it in terms of a hierarchy. Typically cloud storage will be at a lower level with cloud backup layered on top of it. There could also be something else layered on top of it, such as file sync-and-share.
When it comes to using the cloud for backup, it can be accessed in a couple different ways. Backup software can be used to send data to the cloud as a backup target, or backup software from the cloud provider can be used to create restores. Some cloud backup products only work with a particular tool. Granted, some cloud storage only works with certain tools, but generally speaking, if I have Amazon Simple Storage Service (S3) storage, I can access that right from my desktop using a variety of different backup tools or file sync-and-share tools.
Clouds are architected differently for different applications as well. Some cloud storage is architected or designed for general-purpose storage. Some clouds are architected for performance and data is accessed from within the cloud, while some are architected for deep, dark, cold archiving. For example, Amazon Glacier is for archiving, Amazon S3 is for general document and general object serving and Amazon Elastic Block Storage is regular performance storage, but accessed from within the cloud.
A cloud should allow access to its storage from outside, but it should also allow applications to run in the cloud and use that same storage. Think of it as storage with different personalities. Basic cloud storage is just that: storage. Cloud for backup is storage with a backup personality. Or if we were looking at cloud archive -- that would be storage with an archiving personality.
Cloud backup is an application. It's one of a variety of uses for cloud storage. To avoid confusion, you should focus on the functionality that you're getting.
Six steps to a successful cloud DR plan
Cloud-based VMware backup options