- API = Application Programming Interface
- SDK = Software Development Kit
So the real difference is, that an API is no more or less than an interface to "some service", while an SDK is a set of tools/components/classes for a specific purpose. An SDK in fact presents you with an API to interface with. But you might use an API without having the underlying components, for example when the API is provided through a web service.
Examples of APIs:
Examples of SDKs: