It's a question of emphasis, I should say. Perhaps I'm getting too "psychological" here, but in "the mother grabbed her son's arm" the writer's mind is focused simply on the action of grabbing the arm. On the other hand, in "the mother grabbed her son by the arm" they are focused more on the idea of getting hold of the child to stop him from falling. And how does she do it? By grabbing him by the arm.
Not a teacher