Just looking at Softlayer's dedicated machines, it seems they offer a better deal on these than if you go with their Cloud computing offerings. What is the advantage to using clouds/VPS then?
Suppose you have a website that has completely constant traffic. You can therefore anticipate exactly how much computing power you need, and a dedicated machine or two will suit your needs well.
Now suppose that same site (hosted on two dedicated machines) gets a sudden increase in traffic, beyond the capabilities of your two machines. Instead of reserving another dedicated machines, spool up a cloud instance (or two) to help handle the traffic, and once the traffic returns to normal, release the cloud instance(s).
Here's another possibility. Suppose you need 100 machines to crunch data for an hour. Launch 100 cloud server for that hour at their rate and you pay 100 x $0.15 = $10.50 for the hour instead of 100 x whatever they'll charge for a month of dedicated machine usage.
As pedoh said, the advantage of cloud options is the ability to scale flexibly.
VPS, however, is actually more aligned to dedicated servers. Sure, because their hardware limits are manually imposed by virtualisation software, you can scale them, but generally that's designed for when you decide a server upgrade/downgrade is required, rather than on an ongoing basis to meet fluxtuating needs.
Essentially, the key difference between VPS and dedi is that VPS offers a cheaper solution if you want the flexibility of your own server without the cost of hardware that you don't need - as you can get cheap VPSs that give you full root access for $15/20 a month, a deal you'll never find with any dedicated server, regardless of its specifications.
(For my company's websites, we have ~20 quad core dedicated servers because we are able to handle a steady growth of traffic without the need of cloud servers, and we want complete control over them in a way that we wouldn't have with VPS servers. I have 5 VPS servers for personal use however, which lets me spread their power across four data centres in two continents, with various different uses for each of them, without the bill that a similar dedicated server setup would cost, as I don't really need huge amounts of CPU/RAM in each)
One issue I would caution about VPSes is that because there can be more than one VPS on a physical server, there has to be contention for resources. The vast majority of the time this may not (and hopefully will not) be an issue, but imagine that one VPS starts hammering disk. The disk has a performance limit, and therefore the resource can be starved. The managing server will do its best to prevent one VPS from affecting another, but it can and does happen. If the physical server has a huge RAID array of disks this problem will be lessened greatly; likewise if it has 8 cores instead of one. If it's not standard practice for a hosting provider to tell customers what hardware your VPS is running on and how many other VPSes can be on the machine, it should be.
While a VPS or a Cloud solution may provide you greater flexibility for the reasons explained by pedoh, in my personal experience dedicated servers are better served in terms of bandwidth and throughput stability. At the current state-of-the-art however, I'd choose VPS over dedicated anytime (as long as it's cheaper).
We're going to do the same thing. Of course, that "dedicated" server will probably be on site and be multi-use. I'll probably actually just virtualize my production environment within it. It's not so much the virtual / dedicated issue behind that; it's the desire to control my backups. (Although linode's new backup feature seems pretty brain dead simple).
Now suppose that same site (hosted on two dedicated machines) gets a sudden increase in traffic, beyond the capabilities of your two machines. Instead of reserving another dedicated machines, spool up a cloud instance (or two) to help handle the traffic, and once the traffic returns to normal, release the cloud instance(s).
Here's another possibility. Suppose you need 100 machines to crunch data for an hour. Launch 100 cloud server for that hour at their rate and you pay 100 x $0.15 = $10.50 for the hour instead of 100 x whatever they'll charge for a month of dedicated machine usage.