Our eyes don't see frames or scan lines
Wrong, we do. Light (and thus vision) is literally perceived as light waves enters our eyes, once their whole frequency wave goes through from start to end we call that a cycle, hertz are the amount of light wave cycles in a second, and the human eye is equipped to perceive around 1000hz on average (some more, some less). When changes happens at a faster pace, they flicker out or we just perceive them partially, as there is not enough time to process all the light waves or the full extent of the frequencies.
Anything below this threshold can and is in practice perceived smoother, to an extent. The brain becomes the main bottleneck and it can and will ignore received information depending on age, training, genetics, angle (our side vision is usually twice more sensitive), depth, contrast and patterning.
An average person not-teen will not notice much difference when going past 120hz in average PC using situations on their central detailed view. It is also known there is quick diminishing returns when going past 60hz.
Cinema industry did detailed vision studies to identify the lowest refresh rate needed to perceive motion as real (less feels choppy), landed with 23.96FPS and stuck with it. Variable frame rate media adopted 30FPS to avoid dropping below 24FPS. This is literally the minimum for cost saving reasons.
Input lag is a whole different can of issues, and largely a problem created by LCD displays and budget modern era trackpads, keyboards and mouses, and by wireless. 60hz vs 120hz makes frame time move from 12ms to 6ms. Considering the average reaction time (visual identification into action) is of 250ms, and even the most genetically blessed and trained people cannot surpass 100ms, those 6ms gained matters a bit, but not a lot alone. Cheap LCDs (bad response time to signal), older LCDs (all fluorescent and most early LED ones), wireless mouse/keyboards, or wired but bad quality, Windows shenanigans, GPU shenanigans, can easily add a whopping 100ms delay total, and online gaming will add your ping as ms delay too, so things can sum up when combined.
As for 60hz vs 120hz if worth it, kinda personal preference. I have screens of different refresh rates (60, 90, 120, 144), and I'd say that between 60 vs 120, the last one feels "more natural" and "about 30% more smooth" that "suits a bit more nicely", but it really does not matter much unless you become too fixated on the fact (which if you do constantly, like in playing FPS, pick 120hz definitively, then), I prefer visual quality or budget as long as I can keep 60 FPS stable myself.