I found this:
Basically, my take:
Web Apps = Web Sites (just a name change)
Mobile App = new type of app (supposedly same as mobile services but better)
API App = new type of app in App Services
Mobile Services will continue to be supported as standalone service.
I've always thought Azure Mobile Services should not have been called 'mobile' it's an integration of a few Azure Services (Web, SQL, Notification, API, Auth) in the portal which created a nice cloud API backend over data - irrespective of form factor / device.
I see App Services as an improvement of that same concept (minus the 'mobile' name) plus the metadata which exposes these parts in a gallery format.
The trouble with the transition will be the messaging that's saying Mobile App = new Mobile Services. I see it more like App Services = new Mobile Services.