Very simply, Salt API is a REST interface wrapped around Salt. But that doesn't tell the whole story. The salt command is really just a command-line interface for Salt. In fact, each of the other Salt commands (
salt-cloud, and so on) are really just a way to access various parts of Salt from the command line.
Salt API provides a way to access Salt from a different interface: HTTP (or, preferably, HTTPS). Because web protocols are so ubiquitous, Salt API allows software, written in any language that has the capability of interacting with web servers, to take advantage of it.