I was looking through various APIs and SDKs, when I realized that I couldn't really tell the difference between something called an API and something called an SDK.

Both of them are, conceptually, a way for your program to interface with and control the resources provided by another piece of software, whether that other software is a web service, an end-user app, an OS service or daemon, or a kernel device driver.

So, what is the semantic difference between an SDK and an API?

I think it rather falls along the lines of "All SDKs are APIs but not all APIs are SDKs". 

An SDK seems to be a complete set of APIs that allow you to perform most any action you would need to for creating applications. In addition an SDK may include other tools for developing for the platform/item that it is for.

An API on the other hand is just a series of related methods that may be good for a specific purpose. 

As an example, the JDK (Java Development Kit) contains the API as well as the compilers, runtimes, and other miscellaneous tools. The Java API is simply all the libraries that make up the core language that you can work with out of the box. 
