What exactly causes the problems from a rotator cuff injury? muscle weakness or damage?

Rotator cuff tear: A tear in the rotator cuff tendon in the shoulder can cause weakness and pain, especially without overhead activity.

Both: The problems with a rotator cuff injury involve pain, swelling and limited motion. The treatment may involve ice, rest and nsaids. Sometimes a cortisone injection can be helpful. If the rotator cuff is totally torn, one might have limited motion. Then one might need surgical treatment for the problem.

Cuff injury: The rotator cuff is a group of tendons that insert into the upper humerus. The muscles originate on the scapula. A cuff injury is a tendon injury. If describing a cuff tear, it technically is a tendon detachment from the upper humerus.