Azure Cloud Service Role is classic, old style, one of those you won’t see updated counterpart in new Azure Resource Manager (ARM). In Azure, a Cloud Service Role is a collection of managed, load-balanced, Platform-as-a-Service (PaaS) virtual machines that work together to perform common tasks. Cloud Service Roles are managed by Azure fabric controller and provide the ultimate combination of scalability, control, and customization.
Web Role and Worker Role are under Cloud services (classic).
Web Role is a Cloud Service role in Azure that is configured and customized to run web applications developed on programming languages / technologies that are supported by Internet Information Services (IIS), such as ASP.NET, PHP, Windows Communication Foundation and Fast CGI.
Worker Role is any role in Azure that runs applications and services level tasks, which generally do not require IIS. In Worker Roles, IIS is not installed by default. They are mainly used to perform supporting background processes along with Web Roles and do tasks such as automatically compressing uploaded images, run scripts when something changes in database, get new messages from queue and process and more.
Differences between Web and Worker Roles – The main difference between the two is that a Web Role support and runs Internet Information Services (IIS), while an instance of a Worker Role does not.