Try watching a YouTube video at 30 FPS, then watch another at 60. You'll see the difference, even if it's just a video that you watch passively.
When it comes to games, there is a pretty noticeable gap, even more so than a video, cause you don't just see it, you feel it when you're in control. Not surprising, since well... the framerate is doubled.
In fast games, it's especially important (racing, HnS, FPS...), cause the responsiveness of controls at 60 FPS increases thanks to the increase of fluidity caused by the rendering of double the amount of frames in the same time frame.
In short, 60 FPS is more fluid, more responsive, and more pleasant to the eye.
Obviously the higher the better, up to a certain point where we don't see a difference, and that point varies from person to person. I, for example, can see the difference between 60 and 90, but looks and feels all the same to me past 90.