No, the two use different base classes that serve different purposes.  You can see the difference just by looking at the types, MVC Controller and ApiController.

Here's a great quote from this page:

Note  If you have worked with ASP.NET MVC, then you are already familiar with controllers. They work similarly in Web API, but controllers in Web API derive from the ApiController class instead of Controller class. The first major difference you will notice is that actions on Web API controllers do not return views, they return data.