To muddy the discussion by introducing a precise definition, cloud computing has five essential characteristics (per
NIST Special Publication 800-145):
* On-demand self service: You can add/remove services without human involvement
* Broad network access: You can access it over a broad network (e.g. the Internet)
* Resource pooling: Physical/virtual resources are divvied up among multiple customers
* Rapid elasticity: You can quickly add/remove capacity to meet demand
* Measured service: You pay for what you use
The traditional VPS model meets these requirements, and some of the more sophisticated dedicated server hosts probably do too. So, there's nothing particularly special about "cloud computing," other than its embrace of rapid change. There is no requirement for persistent storage... indeed, Amazon EC2 does not have persistent storage without EBS and/or S3. And if you only have a handful of server instances, you won't necessarily reap the benefits of cloud computing. Adding more servers is the "cloud way" to scale up, not resizing your existing ones. Also, if one of your cloud servers dies for whatever reason (hardware failure, software crash, etc), you deploy a new one and life goes on. The reliability of each component is less important than the reliability of the whole.
So, you probably want a VPS, not a self-healing cluster of cloud servers. That's fine.
Back in the day, it was not unheard of for a few friends to chip in on co-locating a decent server in a datacenter somewhere. This meant they could afford nicer hardware and better connectivity than any of them could individually, in exchange for not having exclusive use of the hardware. This arrangement evolved over the years to become the VPS model: multiple tenants on one physical machine, with better hardware than any of them could afford individually. This, in my opinion, is the major benefit of VPSes over dedicated servers. A well-connected and reliably-powered multi-core rackmount server with four high-speed disks in a RAID 10 configuration is not cheap, and it is probably much more power than you need.
So, with dedicated servers, you quickly whittle that down to a poorly-connected, unreliably-powered single-core desktop machine with a single disk drive to save money. With VPSes, you still have the reliable, redundant, and crazy powerful server, but you have dozens of tenants sharing it. You also have a layer of abstraction that's difficult to do with a dedicated server: self-provisioning, all the OS reinstalls and reboots you could want, ssh-based console, automatic online off-server backups, and you never have to deal with "remote hands." The tradeoff is that my $20/mo Linode has less disk space and RAM than my cellphone... but my cellphone doesn't have fifty people sharing it.
..
Also, as a standard disclaimer:
RAID IS NOT A BACKUP.

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