A torn rotator cuff doesn’t necessarily hurt all the time, but it would hurt with activities that require the rotator cuff such as raising your arm over your head, or lifting heavy things away from your body.
Rotator cuff pain tends to feel like pain down the side of your arm. Often it can even go down to the elbow, and classically it hurts at night, particularly when you roll over.
So it doesn’t hurt all the time, but if your shoulder hurts with use, that’s an indication that it might be time to see a sports medicine doctor.