Sep 13, 2017 06:56 AM|Billy Liu|LINK
A web service is a web-based functionality accessed using the protocols of the web to be used by the web applications.
ASP.NET Web API is a framework that makes it easy to build HTTP services that reach a broad range of clients, including browsers and mobile devices. ASP.NET Web API is an ideal platform for building RESTful applications on the .NET Framework.
It is very similar to ASP.NET MVC since it contains the MVC features such as routing, controllers, action results, filter, model binders, IOC container or dependency injection.
And REST always used to make less data transfers between client and server which makes REST an ideal for using it in mobile apps. Web API supports HTTP protocol thereby it reintroduces the old way of HTTP verbs for communication.
And you could refer to the links below for more information about web service and web api.