What, why or when it is better to choose cshtml vs aspx?
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.
Razor is a view engine for ASP.NET MVC, and also a template engine. Razor code and ASP.NET inline code (code mixed with markup) both get compiled first and get turned into a temporary assembly before being executed. Thus, just like C# and VB.NET both compile to IL which makes them interchangable, Razor and Inline code are both interchangable.
Therefore, it's more a matter of style and interest. I'm more comfortable with razor, rather than ASP.NET inline code, that is, I prefer Razor (cshtml) pages to .aspx pages.
Imagine that you want to get a Human class, and render it. In cshtml files you write:
<div>Name is @Model.Name</div>While in aspx files you write:
<div>Name is <%= Human.Name %></div>As you can see, @ sign of razor makes mixing code and markup much easier.
While the syntax is certainly different between Razor (.cshtml/.vbhtml) and WebForms (.aspx/.ascx), (Razor's being the more concise and modern of the two), nobody has mentioned that while both can be used as View Engines / Templating Engines, traditional ASP.NET Web Forms controls can be used on any .aspx or .ascx files, (even in cohesion with an MVC architecture).
This is relevant in situations where long standing solutions to a problem have been established and packaged into a pluggable component (e.g. a large-file uploading control) and you want to use it in an MVC site. With Razor, you can't do this. However, you can execute all of the same backend-processing that you would use with a traditional ASP.NET architecture with a Web Form view.
Furthermore, ASP.NET web forms views can have Code-Behind files, which allows embedding logic into a separate file that is compiled together with the view. While the software development community is growing to be see tightly coupled architectures and the Smart Client pattern as bad practice, it used to be the main way of doing things and is still very much possible with .aspx/.ascx files. Razor, intentionally, has no such quality.