Q:

What is the advantage of Raw Notifications over a normal Request to a server?

I know where Tile Notification and Toast Notification come in handy, when your app is inactive, but what is the advantage of Raw Notifications over the next line?
WebClient( ).DownloadStringAsync( );

A:

Raw notifications allow you to push custom data on-demand to your application while it's running. WebClient.DownloadStringAsync allows you to pull data from a server.

A:

A raw notification is probably best used alongside a pull request in a lot of cases.  I would tend to use a notification purely to tell a running client that  - that could be anything from a new message has arrived server-side (in which case you might want to send out the message in the notification) to "The whole of the data set on the server has been updated, you'd better come along when you can and get the latest changes".
As mentioned, you can use the raw notification to send out all the required information if it is small enough, but on other occasions you will most likely just be telling the app that it needs to call home for some action.  Your choice would have to be based on the expected size of the payload.  From MSDN

The maximum size of a notification is 1 KB for the header and 3 KB for
  the payload.

