If you are thinking about the context of a machine, then there are almost no advantages. I say almost, because it always takes advantage of the tools offered in these cloud environments, but, in contrast, it is a more expensive (proportionally speaking) environment.
The advantages of environments like Azure will appear in cases where you need to scale your site.
An example would be if you have a machine that will run 100% of the time (24 h / d and 7 d / s). And you can rent another machine that will only be switched on at peak times.
Why would anyone want this? Why not a single machine and more "powerful"?
To explain it is necessary to keep in mind that:
CLOUD (the Azure in the case) === Super managed / controlled environment
If it's a "super managed" environment, I can rent a very small machine, pay less for it, and plug in another machine only when my site has the demand for it.
Just to explain what I mean by "super managed" ... It's because, in environments like Azure, there are several variables that we can configure according to our demand. We can configure bandwidth, disk space, memory, etc. (several other things). And once configured we can create images (copies of the machine) and replicate very easily. And still decide what time to turn on / off each machine. And best of all ... This can be, after configured (of course:]), automated.
And with that, companies, when they find the right equation, start to save money.
In short ... Cloud (Azure, in this case) is the place for us to have an environment that always has 100% resource use. That is, nothing idle, nothing to pay for what is not being used.