A path followed, and a structured body of teaching. Both are routes laid out in advance.
Latin cursus (a running, course), from currere (to run); a path already laid out to be followed, extended to a structured body of teaching laid out the same way.
Latin currere is to run, and both senses of the word stay, at the root, about running a laid path rather than wandering one — a river's course and a course of study share the same fixed channel. Neither sense of the word tolerates much improvisation along the way.