When I first getting started on Microsoft Azure, I was confused between these two: Azure Web Site and Web Role.
Azure Web Site is one of many Azure services while Web Role is an instance running behind Azure Cloud Service, another service offered by Azure. Since then, Azure Web Site has been changed to Azure Web App.
Basically, the difference is the control you have and portion you have to manage in each service.
Azure Cloud Service, Web Role in this case, gives you more control, but with more control, comes greater responsibility. See this graph below:
For detail technical difference, see Feature Comparison on Azure App Service, Cloud Services, and Virtual Machines comparison.