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
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).