What are the differences between a muscle sprain and a strain?

Joints can sprain. A muscle "strain" is when muscle fibers get stretched too far and/or some fibers tear. People generally call it a tear if enough fibers break, leading to some bleeding and swelling inside the muscle, resolving later with some healed scar tissue and some tightness in the area (less elastic). A joint sprains when its ligaments (whitish fibers) are stretched too far and break. Muscles don't "sprain".
See below. Muscle fibers are injured that causes muscle strain. Ligaments can be injured and the terminology is ligament sprain.