API: Application Program Interface
This is the set of public types/variables/functions that you expose from your application/library.
In C/C++ this is what you expose in the header files that you ship with the application.
ABI: Application Binary Interface
This is how the compiler builds an application.
It defines things (but is not limited to):
- How parameters are passed to functions (registers/stack).
- Who cleans parameters from the stack (caller/callee).
- Where the return value is placed for return.
- How exceptions propagate.