Continuous spectrum. Causes of muscle stiffness and soreness can be similar, and these terms merely represent dimensions of severity, as stiffness often accompanied by soreness.
Similar but not same. Muscle soreness is often due to overuse of muscles and is likely due to build up of lactic acid in the muscle and will resolve with time and recovery (hydration helps). Muscle stiffness can result from soreness but is often a result of failing to stretch before and after a workout. Stiffness decreases the Range of motion of a muscle or joint and can improve with stretching and/or yoga.