1/60 delay is 0.016s. 1/120 delay is 0.008s. 1/540 delay is 0.0018s.
These numbers are just marketing and self-convincing. And making money from that.
Human body does not work that fast, typical eye to movement reaction time is on the scale of 0.05s-1s.
Basically, you won't notice the difference *unless* you imagine yourself doing it (placebo effect).
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P.S.
The bottom line is, our eye-brain channel and reception/recognition has finite and pretty much measurable 'speed'. That's why we are able to have a 'stroboscope effect' under disco lamps. That's why car quickly rotating wheels suddenly start 'rotating in other direction' visually. That's why quickly rotating fans look like shady circles. That's why we perceive CRTs, LED displays and lamps (especially LED) not as blinking mess but as a stable moving picture or light source. That's why fast (especially bright/contrast) moving objects become lines in our sight. That's why we can watch anime without puking even if it's mostly 5-15 FPS at all. Etc. Etc.
It's non-linear. It all depends on overall light levels, scene contrast, distance & active eye focus, concentration on object, their position and speed of movement, our condition, age and level of fatigue, etc. Hell, even the vision position. Center is 'faster' than 'edges'. Eye sensors need to recharge as well, although they do that in an alternating patterns.
But overall, it's measured to 30-40 FPS at average. The problem with i.e. 30-40 Hz of lighting being 'blinky' is that our vision is also 'framed', and so if lighting up does not match our 'frame', we do see it only partially.
And that's it. Almost any complex motion over 60 FPS is bound not to be different. Except if it's a medium sized object (let's say apple sized) exactly in the center of vision at medium (3-5m) distances. Yes, you can perceive such objects faster. Yes, you can perceive full scene light up and down faster. And there were 'tests' and 'articles' on that. But you can't do this constantly in a minute or more time frame. Or on more than full scene toggling light up and down or a single object moving. Eyes have their limits. Brain has its limits.
Then how 30 to 60 FPS are that smoother, and 120 FPS are a bit even more smoother?
Interpolation, sir.
Eye sensors are 'charging' (or should I say 'depleting') with light hitting them and so they accumulate the effect, much like the photo cameras (yes, exposure time, exactly) do. Thus, changes happening at speed eye can't perceive are averaged together into a single 'frame' in each of sensitive elements of our eye.
Let's talk MSAA, on a good resolution display and appropriate distance.
No MSAA to 2x MSAA is usually hugely different and better, yes? That's our '30 FPS' (on the average of vision speed). Yes, these 30->60 FPS is that 'MSAA 2x' for the vision speed that makes moving things much smoother, like MSAA 2x does for the subpixel resolution that cannot be rendered.
Now take 60->120 FPS. That's MSAA 2x->4x. Yes, some will notice the difference. But many will tell it's not worth the hassle and extra resources or it gives nothing.
Now take MSAA 4x->8x... You know... No difference. Almost. And that's what happens with FPS bloat as well. The more we bloat, the less differences there are.
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P.P.S.
Personally, I'd appreciate 8K more than any of >120 FPS on commonly available displays. And yes, better color calibration. That's because photos do look like awry on poorly calibrated displays. And because 4K dots are still distinctly perceivable, so not even close to the eye limit even on 32". While 120 FPS is already times over the top.
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P.P.P.S.
What I wrote is very simplistic aggregation of the principles. It's not exactly correct if we go into detail, it's just the layman basics. If one wants, one can study more on the matter - and find different proofs and some divergencies to that. Or one can even just believe 240-1000 FPS will magically work improving one's reaction time, what's interesting here is that if this belief is strong and self-convincing enough, it will.

Just 'cause brain likes stimulus. So yeah, it may still work for some even if it works in a bit different way.