The only place I could possibly imagine that mattering is for fast-twitch video games, or theoretically videos (though most of those are still 24 or 30 fps).
And framerate in UI is a different beast compared to video. A menu animation running at 24fps feels much choppier and sluggish than a video running at 24fps.
I really don't want to argue about this so I figured I just let you experience it yourself: http://jsbin.com/qizatugepo/1/edit?js,output
Frame drops are a different story entirely. The swap_control_tear extension for OpenGL takes care of that to some degree though.
Having said that, shooting for your monitors refresh rate isn't a bad idea — it's just that you could use those cycles (and watts) for something else. And even worse, if your app is running at 60 fps most of the time but drops to 30 fps every now and then, then it's gonna look a lot worse than the one that runs at 30 fps consistently.
It's about scrolling and animations. In videos it 24fps is enough because you have motion blur. In fact higher frame rate makes videos seem unnatural..
>Do people really care if their website/mobile app is 60 fps? Who are these people?
People who prefer native apps to webapps. Quite a lot of them actually.
It's mostly related to the animations. I.e. scrolling, dragging. This is what makes the experience amazing.
But also, it's because every time you do a costly operation, ti freezes the screen and the user loses control. This is a solved problems in so many environment, frustrating that it's not there yet while developing cordova apps.