I have created an Azure Web App today and it has a Mobile section in the Settings which contains Push notification, Mobile Authentication, etc. You can even download the source
I have found some subtle, but key, differences. Assuming a C# backend, the Mobile App Service controllers will inherit from TableController - which provides out-of-the-box CRUD operations on a table, compared to the Web Api 2 style ApiController (which you can still use in a Mobile App Service). Any less confused now?