What is the difference between an API and a Web Service? Are they the
same thing? For example, is the Google Maps API a web service?

If the system you are interfacing is designed to allow a computer to
access it directly, rather than provide a user interface, that's an
API.  If the system is designed for a person to use it, that's a
service.  Most online mapping websites are going to be a service.  An
example of an API would be Ebay as they provide a method for other
software to directly interface without using web pages or user
screens.

It is possible to simulate the actions taken by a user on a website
and thereby treat a website's services and pages as though they are an
API, but if the site changes then your software often breaks.  An API
usually expresses a committment by the provider to continue to provide
access without making changes that might break software designed to
access it.