As other people have answered, .cshtml (or .vbhtml if that's your flavor) provides a handler-mapping to load the MVC engine. The .aspx extension simply loads the aspnet_isapi.dll that performs the compile and serves up web forms. The difference in the handler mapping is simply a method of allowing the two to co-exist on the same server allowing both MVC applications and WebForms applications to live under a common root.
This allows http://www.mydomain.com/MyMVCApplication to be valid and served with MVC rules along with http://www.mydomain.com/MyWebFormsApplication to be valid as a standard web form.
Edit:
As for the difference in the technologies, the MVC (Razor) templating framework is intended to return .Net pages to a more RESTful "web-based" platform of templated views separating the code logic between the model (business/data objects), the view (what the user sees) and the controllers (the connection between the two). The WebForms model (aspx) was an attempt by Microsoft to use complex javascript embedding to simulate a more stateful application similar to a WinForms application complete with events and a page lifecycle that would be capable of retaining its own state from page to page.
The choice to use one or the other is always going to be a contentious one because there are arguments for and against both systems. I for one like the simplicity in the MVC architecture (though routing is anything but simple) and the ease of the Razor syntax. I feel the WebForms architecture is just too heavy to be an effective web platform. That being said, there are a lot of instances where the WebForms framework provides a very succinct and usable model with a rich event structure that is well defined. It all boils down to the needs of the application and the preferences of those building it.