My old 'Web API' point of view ...

A web Application Program Interface, is a set of callable functions or methods that a web application exposes as a public interface for communication with external application.

... and the modern Web Services definition

"A Web service is a collection of functions that are packaged as a single entity and published to the network for use by other programs. Web services are building blocks for creating open distributed systems..." [1]

[1] from "The Web services (r)evolution, Part 1", http://www-106.ibm.com/developerworks/webservices/library/ws-peerl.html