Is there still a difference between a library and an API?

My view is that when I speak of an API, it means only the parts that are exposed to the programmer. If I speak of a 'library' then I also mean everything that is working "under the hood", though part of the library nevertheless.

A library contains re-usable chunks of code (a software program).

These re-usable codes of library is linked to your program through APIs (Application Programming Interfaces). That is, this API is an interface to library through which re-usable codes are linked to your application program.
In simple term it can be said that an API is an interface between two software programs which facilitates the interaction between them.

For example, in procedural languages like C, the library math.c contains the implementations of mathematical function, such as sqrt, exp, log etc. It contains the definition of all these functions.

These function can be referenced by using the API math.h which describes and prescribes the expected behavior.

That being said, an API is a specification (math.h explains about all the functions it provides, their arguments and data they return etc.) and a library is an implementation (math.c contains all the definitions of these functions).

I think that Library is a set of all classes and functions that can be used from our code to do our task easily. But the library can contain some of its private functions for its usage which it does not want to expose.

API is a part of library which is exposed to the user. So whatever documentation we have regarding a library, we call it an API Documentation because it contains only those classes and functions to which we have access.

API is part of library that defines how it will interact with external code. Every library has API, API is sum of all public/exported stuff. Nowadays meaning of API is widened. we might call the way web site/service interact with code as API also. You can also tell that some device has API - the set of commands you can call.

Sometimes this terms can be mixed together. For example you have some server app (like TFS for example). It has API with it, and this API is implemented as a library. But this library is just a middle layer between you and not the one who executes your calls. But if library itself contains all action code then we can't say that this library is API.