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
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.