Whenever I ask people about the difference between an API and a library, I get different opinions. Some give this kind of definition, saying that an API is a spec and a library
we have first to define an interface ...
Interface :is the means by which 2 "things" talk to each other and exchange information. "things" could be a (1) human or (2) a running code of any sort (e.g. library ,desktop application , OS , web service ... etc).
if a human want to talks to a program he need Graphical user interface (GUI) or command line interface (CLI). both are types of interfaces that humans (but not programs) would like to use.
if however a running code (of any sort) want to talk to another running code (of any sort) it doesn't need or want a GUI or CLI ,it rather need an Application Programming Interface (API).
so to answer the original poster question : library is a type of running code and the API is the means by which this running code talk to other running codes.
According to my perspective, whatever the functions are accessible to invoker , we can called as api in library file, library file having some of the functions which is private , we cannot access them ..