What's wrong with liking 60FPS?
Or going to the point, higher FPS, say 200FPS?
I mean, sure it will always be smoother, the human eye can even get small reactions/hints up to 500FPS (like a short flashing frame), so what is wrong about longing for that?
It is the same as saying "why do people want more polycount in their game models?", you know (given the model is well made) more polycount means more detail, the same as "why do people want a bigger screen resolution?" or "why do people want higher quality textures?". This also makes the visual quality better (in a independent direction from FPS).
So then comes the obvious problem here, you could say the visual quality of moving pictures is composed of fluidity (FPS) and picture quality (depending on resolution, polycount, shaders, etc).
Both fluidity and picture quality compete for limited resources, so you have to make a tradeoff. If your reactions and inputs influence the outcome of what is being shown (as in eg. a video game) fluidity would be way more important than in something which outcome doesn't depend on your reactions (say a film). If you improve fluidity you reduce your own lag and you improve the gaming experience.
Out of my preferences I would say you have to go for the required acceptable FPS for the type of game, and only then improve the picture quality as much as you can, if you then can get extra FPS then that is a big plus.
Anyone that tells you the human eye can only see 24FPS, or 60FPS, or 200FPS, well, this person is lying. Those FPS were set due to technological restrictions, it was a tradeoff, and the human eyes and brain are good at adapting to them and filling the blanks, but humans can see and react to way shorter stimuli, even short enough that your brain is not able to process them completely but your "reflexes" can react to them.
That said, of course you can go out of your way and drop resources like crazy to, let's say, make your daily conmute on an F-22, sure that would go fast, but is it worth it? Sometimes it is better to just take your standard cheap domestic flight service, it is good enough and it will not drain your pocket. The same thing applies to the diminishing returns of improving an old game engine or buying additional hardware.