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

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 is an implementation...

Some will tell you this type of definition, that an API is a bunch of mapped out functions, and a Library is just the distribution in compiled form.

All this makes me wonder, in a world of web code, frameworks and open-source, is there really a practical difference anymore? Could a library like jQuery or cURL crossover into the definition of an API?

Also, do frameworks cross over into this category at all? Is there part of Rails or Zend that could be more "API-like," or "libraryesque"?

Really looking forward to some enlightening thoughts :)

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).

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.