This is a misunderstanding.
Lets split this into two things: can you see the difference, and does it make you a better player.
I moved from 60Hz -> 85Hz -> 144Hz -> 280Hz and each time, I could see the difference. I know everyone on here is a contrarian, so let me expand. 280Hz is just a nicer experience to me: everything is very fluid feeling, there are fewer artifacts.
You can see a difference because your eyes are an additive input device and games can only render instants in time. You can't identify individual frames at that pace, but you will see all of the motion data as a blur that'll be a lot smoother than 60Hz would provide.
Movies get less benefit from high frame rate because a frame doesn't record an instant, it records a period (shutter speed). This encodes a lot of correct motion data into the frames themselves. I think this is part of where the 60Hz myth came from. But, the benefit is still there even then.
Okay, so you can see the difference. But does it make you a better player?
So long as you can manage it without large input latency (some games will try to render ahead -- big no-no), up to about 120Hz I think it does explicitly but only slightly make you a better player. Beyond that, I bet some esports pros could make some use of it but for normal people, it's all about the smoothness quality.
It really does provide a benefit, as a lifelong gamer there's been nothing quite like switching to modern OLED which has input latency that rivals or beats CRTs, framerates that rival LCDs without smearing, and image quality that beats them all.