Coarse means “common,” “rough,” or “vulgar.” Course implies movement over time or over a predetermined route. The phrases “in due course,” “matter of course,” and “of course” are all built on this meaning. A course is also a subject you take in school.
The word course also has transitive and intransitive verb meanings. It means “to move swiftly.”
Blood coursed through her veins (intransitive). The ships coursed the seas (transitive). Example: The coarse material on the seat made biking the torturous course that much more unbearable.