You cannot see a difference, but you can feel it. If you played for a long time at 240hz then went back to 120hz you would feel a difference.
Human eyes cannot see above even 20 and that's a fact I've read from an eye doctor/optometrist. But you DO notice the movement differences.
24fps movies would flash the same image 2-3 times to create a 48-72hz video that played on how afterimages work in your eyes/brain. Higher framerates numbers fool the brain into thinking that a bunch of frames are a moving scene, and the higher the framerate the smoother it looks. In the case of video games they dont flash the same frame multiple times at lower framerates so things stutter more.