Well, the main pro for AWS is that you have total control over the server which means that you are not going to have problems installing system dependencies, whereas with RStudio Cloud you don't have administrative access to the server.
Also, by having control over the server you can install other tools on the same virtual machine that usually play well together like Shiny Server, a SQL server, Nginx, LibreOffice, Tex Live, etc.
The con is that you have to install, configure and maintain all this by yourself.