Updating the answer to current state of Azure,
App Services now replaces all Mobile, Api and Web Apps flavors as a single app framework with all the functionality rolled over to make things more accessible across application types. Currently all of Web, Mobile and Api Apps are collectively called App Services. We still offer customer to be able to create a Mobile App and a Web App in the gallery but that is basically resolve into an App Service App.
https://azure.microsoft.com/en-us/documentation/articles/app-service-api-apps-why-best-platform/
Features for Mobile work for Web App as well such as Easy Tables and Easy API. And features for API apps like API Cors and API definitions now work on web apps as well. A customer can host a single web app to act as any mobile service or an api with all the features offered through the app services.
We also have a new service in preview particularly targeting API Apps by offering a management experience for your APIs, Basically you can control the generate try API pages, gather execution analytics, throttle and much more. Check out the feature blog to learn more about the Azure API Management Features. And yes you can host the APIs as a App Service App and hook things up with API Management.
https://azure.microsoft.com/en-us/documentation/articles/api-management-get-started/
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