You've seen games running at 120Hz and at 60Hz. The difference is obvious, isn't it? The difference between 24Hz and 60Hz is certainly obvious: that's the visual difference between movies and TV sitcoms.
I can type about 90 words per minute on QWERTY, which is about 8 keystrokes per second. That means that the average interval between keystrokes is about 120 milliseconds, already significantly less than my 200-millisecond reaction time, and many keystrokes are closer together than that—but I rarely make typographical errors. Fast typists can hit 150 words per minute. Performing musicians consistently nail note timing to within about 40 milliseconds. So it turns out that people do routinely time their physical movements a lot more precisely than their reaction time. Their jitter is much lower than their latency, a phenomenon you are surely familiar with in other contexts, such as netcode for games.
If someone's latency is 200 milliseconds but its jitter (measured as standard deviation) is 10 milliseconds, then reducing the frame latency from a worst-case 16.7 milliseconds (or 33.3 milliseconds in your 30Hz example) to a worst-case 8.3 milliseconds, and average-case 8.3 milliseconds to average-case 4.2 milliseconds, you're knocking off a whole 0.42 standard deviations off their latency. If they're playing against someone else with the same latency, that 0.42σ advantage is very significant! I think they'll win almost 61% of the time, but I'm not sure of my statistics†.
See also https://danluu.com/input-lag/#appendix-why-measure-latency:
> Latency matters! For very simple tasks, people can perceive latencies down to 2 ms or less. Moreover, increasing latency is not only noticeable to users, it causes users to execute simple tasks less accurately. If you want a visual demonstration of what latency looks like and you don’t have a super-fast old computer lying around, check out this MSR demo on touchscreen latency.
> The most commonly cited document on response time is the nielsen group[sic] article on response times, which claims that latncies[sic] below 100ms feel equivalent and perceived[sic] as instantaneous. One easy way to see that this is false is to go into your terminal and try sleep 0; echo "pong" vs. sleep 0.1; echo "test" (or for that matter, try playing an old game that doesn't have latency compensation, like quake 1, with 100 ms ping, or even 30 ms ping, or try typing in a terminal with 30 ms ping). For more info on this and other latency fallacies, see this document on common misconceptions about latency.
(The original contains several links substantiating those claims.)
https://danluu.com/keyboard-latency/#appendix-counter-argume... has a longer explanation.
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† First I tried sum(rnorm(100000) σ of latency being added; it's a random latency of up to 0.83σ, so I tried sum(rnorm(100000) rnorm(100000) + runif(100000, max=0.83))/1000, we get more like a 60.8% chance that the 120fps player will out-twitch them. I'm sure someone who actually knows statistics can tell me the correct way to model this to get the right answer in closed form, but I'm not sure I could tell the correct closed-form formula from an incorrect one, so I resorted to brute force.