Full FUNCTIONALITY of the system is CONSIDERED in API testing as it will be used by the end-user (EXTERNAL developers who will use your API )
API test are often run after the build is ready and authors do not have access to the source code.
UNIT testing:
Unit testing is owned by development team
Unit testing is white box testing
Unit testing is done to verify whether each unit in isolation performs as expected or not
For each of their module the developers are expected to build unit tests for each of their code modules and have to ensure that each module pass unit test before the code is included in a build.