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
Like you, I couldn't get my head around the difference between the different types of App Service apps (Web Apps / Mobile Apps / API Apps). Judging by the fact that no one has really answered your question, it looks like we weren't the only ones.
It made no sense to me that Web Apps, Mobile Apps and API apps were distinct things: surely the whole point is that you should be able to expose an API that serves your web site, your mobile apps, any other kind of client, making use of whichever features you need for your use case?
And, lo and behold, buried deep in the Azure docs:
The only difference between the three app types (API, web, mobile) is the name and icon used for them in the Azure portal.
Yep: they're exactly the same.
Mobile Service is the "classic" version and is only available in the classic portal. Mobile Apps is part of App Service is only available in the preview portal.
App Service includes Web Apps, Mobile Apps, API Apps and Logic Apps is deployed as one unit. Using the classic way, you would have to create and pay let's say for a Web App and a Mobile Service so you can save money using the App Service. You also get a larger selection of VMs.
Not all Mobile Services features have been migrated yet to Mobile Apps. If you use a Node.js backend, there's no nice UI built-in in the new portal to manage your JS code.
Note that Microsoft has not announced that it will deprecate Mobile Services.
In old days of Azure, they had 2 offerings - Azure Websites and Azure Mobile services. Some time ago, both of them got rolled into something called Azure App Service which also included two more kinds of apps - API Apps and Logic Apps.
To answer your question specifically, Azure Web App is new name for Azure Websites that you can use to build websites in multiple languages where as Azure Mobile App is the new name for Azure Mobile Service that provides backend infrastructure for mobile applications (or in other words, Azure Mobile App is Mobile Backend-as-a-Service [MBaaS] offering from Azure).
Sorry for the late reply. They all are really the same thing. If you look in Azure Resource Explorer (available in the application menu of your app or resources.azure.com) you will see the indication of how we display a different icon for it - "kind": "mobileapp", for Mobile Apps for example. We leverage the different kinds to provide different quick-starts. You can certainly create an Azure Web App and load Azure Web App code to it to get the Mobile Functionality.
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?