What is the difference between an API App and a Web App?
There are many minor difference between Web API and API Apps, but the very notable and key differences are
Native Swagger implementation - When you create API App in Visual studio, swagger reference comes by default. Swagger provide very developer friendly features for API consumers to Interact with your API thru Swagger UI. Also Swagger based API's provides client SDK generation (both .Net based client and Javascript based client) which makes easy to call API's just like regular method call. Note: Swagger implementation on regular Web API is possible manually.
Ability to publish your API Apps into Azure Market Place. Azure Market Place is the public repository for all API Apps that can be consumed freely or by charge.
this 15 minute video from Channel 9 gives an excellent overview about Api Apps.
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/
To supplement Greg's answer, Here's an even more recent article describing the differences.
To sum up:
"The key features of API Apps – authentication, CORS and API metadata – have moved directly into App Service. With this change, the features are available across Web, Mobile and API Apps. In fact, all three share the same Microsoft.Web/sites resource type in Resource Manager."
And here's another important note:
"If your API is already deployed as a Web App or Mobile App, you do not have to redeploy your app to take advantage of the new features."
There was a point in time when there were differences between the different app service types, but that is no longer true. The documentation now states:
The only difference between the three app types (API, web, mobile) is the name and icon used for them in the Azure portal.
So it no longer matters which app service type you choose to deploy to (unless you care what the icon looks like).
UPDATE
Function apps are now the exception. Creating a function app changes the user interface in the portal. The underlying web app, however, is no different. Setting an app setting named FUNCTIONS_EXTENSION_VERSION = ~1 turns any web app into a function app (minus the user interface in the portal).