Shooter player here. This is a laughable statement. Pretty much anybody can tell the difference between 60hz and 120hz - even the mouse cursor moves smoother. Good players try to play at higher than 120hz rates, because it does make the difference.
If the mouse were moving continuously you would be able to see the full trace of its path and not just a few discrete locations along the way. A higher frequency would get closer to this.
Also, in an interactive scenario, the frame rate will impact how quick you get feedback on your actions. With 60Hz, the frames each take 17ms, meaning the next frame will be on average 8.3 ms away. Thus if you perform an ingame action, you will get visual feedback in ~8.3ms, whereas with 120Hz it will come in 4.1ms. There is probably some kind of buffering done, so likely your action will not end up in the currently drawn frame but the one after that. So the response will on average come 3/2 frames after your action: 25ms later for 60Hz v.s. 12.5 ms for 120Hz. 10ms faster reaction time should be noticeable.