You only notice the difference when you play 60fps over a longer period of time and then you have to go back to frame drop hell. Also you need proper test material. 60fps =/= smooth gameplay. There are certain games where the average fps might be 60 or above but they're still choppy as fuck due to high variance of frame times. That's why a 30fps game can feel smoother than fps between 50-70 because it has more constant frame times. That's not the norm however.
The next thing is that you basically have double the input lag on 30fps. I tried Driveclub and while it was running relatively smooth I had a very hard time due to high input lag. Same with Rocket League which rarely comes close to 60fps.
Also, it doesn't stop at 60fps. That's just an arbitrary line due to traditional display refresh rates. More is always better. People who have played for a long time on PC with powerful GPUs are used to 100 or more frames and will be able to tell the difference between that and and 60 fps. 60fps doesn't mean infinitely smooth.
Another new factor is G-Sync/freesync which basically removes choppy frames caused by the display refresh rate, smoothing the picture even further without the issue of tearing.
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