1. Want to resize your server? you can do it. 2. Want to create a new server and move some kind of elastic IP to point to this new server? you can do it with most cloud providers 3. Want to backup the whole disk, very easy to do it. 4. Want to spin up a 1000 instances for an hour? cloud servers can do it. 5. Want hourly pricing? cloud servers can do this.
1. dedicated servers are capital expenses. 2. cloud servers are operational expenses, which is usually preferred.
If terms of actual operations I do not really see a huge difference. Both require a decent amount of preparation, tooling these days is awesome and you can have your own cloud servers with ease and enjoy the same tooling.
E.g. https://www.terraform.io/docs/providers/openstack/index.html
The biggest benefit of cloud servers: if you have no clear view what your project _will_ need, this is the place for experiments.
The biggest benefit of dedicated servers: being in control. You control the uptime, and thus SLA fulfilment.
Biggest danger of cloud servers: paying too much because you have no picture of what work loads will incur which costs.
Biggest danger of dedicated servers: overdoing it, when all you need is a simple machine.
1. Upfront ownership costs versus hardware rental costs.
2. Performance optimization versus versatility optimization.
3. Maximum administration capability versus maximum maintainability and reliability.
I mention these three pairs of axes because with the notable exception of ultra-low network latency and the ability to quickly expand or diminish computing power (both of which fall under 2), you can reasonably achieve everything in either a cloud computing or dedicated hardware environment.
If you have a lot of upfront cash and you're intimately familiar with your long term computational needs and upgrade requirements, it probably makes more sense to use dedicated hardware. This generally doesn't describe young companies. You can amortize the cost of the hardware over the lifespan of its use and you'll typically find it's far cheaper (orders of magnitude) to run a dedicated server on a 1:1 cost basis.
If you aren't aware of what your computational needs will be within the expected lifespan of new hardware, it is safer to abstract your resources to a cloud environment. Naturally this will be more expensive over the long term, but it will hurt less if you're e.g. just trying to build a product and validate it without committing to large capital requirements.
From there it depends on use case. If you have performance requirements that restrict the amount of network latency you can tolerate, dedicated hardware is obviously better and more predictable. You can colocate it if need be, though this comes close to cloud computing unless you actually own the hardware and merely use a datacenter for the networking advantages. If you have reliability requirements that can tolerate reasonable network latency, cloud computing is better for redundancy.
Theoretically you can even buy much larger dedicated computing capacity than you need, virtualize it in whichever custom configuration you want, and simulate the "burstability" of cloud providers. But you won't get the regional redundancy this way.
The last major difference is maintainability. If you operate your own dedicated hardware, you become a sysadmin for it. This can be very empowering, but it can also be a huge time suck, especially for companies that don't need that sort of customization capability.
There's otherwise no inherent difference - the cloud is just someone else's computer after all. Generally speaking any business use case can be thrown onto either platform, and you'll want to do a cost benefit analysis on which will be more stable, align more closely with your needs and satisfy budget requirements. Personally I own and operate what could be considered a small supercomputer in my home, and the only thing I use cloud resources for is redundant backups.