Firstly, "unlimited bandwidth" doesn't exist. There will be a fair use policy that applies. Secondly, what you are buying into with Amazon's cloud service is their infrastructure that is proved to be able to deal with load.
Secondly, a cloud provider like EC2 or Azure gives you the ability to scale your application to multiple servers very quickly if needed, and scale down when demand is lower. For example, if you were buying dedicated servers to deal with a single peak period that you have in the year, for the rest of the year the potential of those servers would be sitting dormant. With a scaling cloud service you only pay for the time that the additional servers are required.
Finally, depending on what software you are looking to run on the server there may be licensing implications that will be your issue if you use a dedicated server, that will not be there if you use a cloud service.
While the upfront cost might look similar, or even cheaper for a dedicated machine, you should consider the TCO.