APIs are naturally part of SDKs. It is how you interface with code in libraries (jars, dlls, etc). When I worked on a library/package, I just had to write functional tests that used the library/package.
SDKs however are installable packages so there is also testing that goes along with that. When I worked on an installable toolbar we had to test the installation against many different Operating Systems. We also had to test it against many anti-virus vendors to address false-positives. There's also testing if the install is interrupted, or checking the background process for memory leaks.