[Modified to answer the question posed in the title]
As an experienced developer, you would use the Web Pages framework when you want to get something done quickly (POC, for example), or if you want to teach someone how to get started with ASP.NET, perhaps.
You are asking two separate questions here: Web Forms View Engine vs Razor View Engine in MVC, and Web Pages Framework vs Web Forms Framework.
As others have said the first decision will be down to preference. Neither does anything that the other doesn't. It's a bit like the C# or VB debate.
The Web Pages framework is intended to provide an easier "in" for beginners than Web Forms or MVC. It can be considered a very belated replacement for the much easier classic ASP framework and it competes with PHP in that respect. If you need separation, testability and so on you should look to the MVC framework. The Web Pages framework doesn't offer very much at all in terms of enterprise development approaches.are used to Web Forms you are unlikely to gain anything from Web Pages. Separation is a lot more difficult. It doesn't encourage layering of code or testing. No server controls or code-behind files. And Web Pages only offers the Web Site deployment (compile on first run) as opposed to the Web Application which can be pre-compiled.
Razor itself was developed for the Web Pages framework. MVC liked it and leveraged it as a new view engine option. But essentially, you have three ASP.NET frameworks to choose from now: Web Pages, Web Forms and MVC.