Just for fun, I tried playing the original Super Mario Bros at 30fps. Even with that game, the difference was noticeable. Jumps were harder to time, and Mario's control felt more slippery. It matters for pretty much every game out there unless it's turn-based.

The difference is noticeable to me.
Any good gamer will be able to tell the difference right away as it influences things like reaction times/responsiveness. In fighters every move has a number of frames for example. Every frame can make the difference.
Every game benefits from it from a visual standpoint as every animation will be twice as smooth.

For a casual gamer that doesnt even try to execute well, yeah maybe they don't even bother to try and notice it. But i can tell that whoever says that 60fps doesnt matter, doesnt put much effort on their gaming. Because its a huge thing. If you honestly don't see the difference i can only conclude its a physical problem.

I've played lots of racing and FPS at 30fps and is ok for me. I won't question the fact that 60 could give a better experience but i think 30 is ok if the game is well designed with that in mind.

True, it depends on how the game is programmed. But if I had to bet, I'd bet on the 60 FPS having less input lag. And I fully agree with your last statement.

I can notice the difference between 60 fps when going straight from one to the other, but after like 5 minutes you get used to it.

I don't think fps is alpha and omega. As long as it's decent I don't really care.

It's not made much of difference for me, but it seems that the games people list for where it matters are games I don't really play much, racing, fps and fighting

Platformers, shooters and racing games should be on 60fps. But TPS like Uncharted, action-RPGs like Dark Souls... can do well enough with 30fps. I think it depends on the game.

Yes, it changes the experience a lot

Depends on the game. I have Dragon Age Inquisition locked to 30 fps just because I don't notice 60 fps much in that game, and I have to turn down the settings quite a bit to get a constant 60 fps. In Fallout 4 I notice a huge difference and can tell without there even being a comparison which is 60 fps and which is 30 fps. Most first person games or games that have fast camera movement are the same deal. If the camera is static I don't really mind 30 fps.  The thing though is a drop below 30 fps is horrible, while a drop below 60 fps to say 55 fps or 50 fps doesn't bother me too much if it isn't frequent. 

Play Destiny for an hour and then go to COD. The difference is jarring to your eyes.