One huge benefit that hosting videos in a cloud infrastructure brings is that you're only paying for what you use, and you don't have to worry about upgrading drives and expanding space constantly. With user-generated content, this might quickly become burdensome and so an elastic solution is way better. Mechbunny recently introduced integration with MojoCloud's S3-compatible Object Storage - definitely something to check out for everyone using MB.
Cloud isn't necessarily cheaper than regular dedicated servers. If you look at Amazon's prices, they consist of a number of variables that, per my experience, tally up to a pretty significant amount, often more than what you would spend with a server hosting the same resources. At MojoHost, we strive to limit the number of variables the customer pays for, making the pricing structure more straight-forward.
This all said, there technically is no cloud. It's still a bunch of servers that have your content, distribute and serve it to your customers. It's just way more clever about redundancy and faster, but there's still physical hardware behind the "Cloud"
Per my limited understanding, cloud is the abstraction of CPU, network, and disk. So, you have a pool of resources provided by a lot of different physical machines, which you can carve up and distribute between your (or in case of public cloud your and other clients' needs) and use as needed.