Frame-rates in film and games are two very different issues. Not only because film doesn't have to worry about input delay, but because 24fps has been the standard for so long that people have become accustomed to the visual 'feel' of it. Hence many thought the Hobbit looked weird in 48fps compared to the 24fps version, but few would say a game looks weird in 60fps. When someone says they prefer 24fps in film, they usually just outright prefer it. When someone says they prefer 30fps in a game, it's (almost always) because they like the positives more than the negatives that cut provides.

Ever since I played a 60 FPS game, and tried playing a 30 FPS game, my eyes felt extremely weird.

I'd say 60 FPS is very important, though I'd say games like traditional RPGs can get away without having 60 FPS, as there isn't a whole lot of movement in them.

Even 60 fps games can have quite a bit of input lag, but it'd still probably be less than a game at 30 fps. But what's definitely the worst is when developers feel the need to cram every bit of detail into a game so it looks pretty in screenshots and video, but then it can't even maintain a stable 30 fps as a result of all the detail. (I understand that this is primarily an issue on consoles, and not so much on PC)

It doesn't matter to me as long as the frame rate is stable. Yet I hardly notice when a game drops from 60 to 40, a 30 to 20 drop is much more noticeable. Micro stutters stand out more in 60 fps games though.
I did notice some slowdown on Fallout 4 last night, didn't matter to me. I wouldn't want the game to drop to 720p for better frame rate. It's more important to be able to see what's coming for me from the distance.

There are many factors involved of course. If the game adds heavy full screen motion blur and temporal AA than 60fps is definitely preferred to limit the amount the image is blurred by the effects. Adding full screen notion blur is imo the worst solution to 'improve' perceived frame rate.

It also depends on your max turn rate. On PC you can turn as fast as you want with mouse look, the faster you turn the higher the fps you need to keep the image stable. There are strict guidelines for movies for how fast the camera can pan at 24 fps. Which is also why 60 fps is more important for fast sidescrollers and 2D platformers.

There are 2 things involved for perceiving fluid motion, besides frame rate, the size of the steps things take across your visual field also play an big factor. Which is another reason why it's more important for traditional PC gaming where you sit closely in front of a monitor. The bigger fov the image takes up, the bigger the steps are at which things jump across your visual field. And conversely, flying straight high above the landscape in flight simulators looks perfectly fluid at 20 fps.

Racing is my favorite genre and I have played well over a hundred of them through the years at all kinds of frame rates. Maybe not a popular view, yet for sim racers 30 fps is enough for me. You don't turn that fast, and it's more important to have the best view of the track ahead.
For fast arcade racers 60 fps is better with unpredictable movement, spinning out, crashing and all kinds of stuff flying at you. DriveClub is an interesting case. For the cars 30 fps is fine. Yet with the bikes it's a bit low in first person mode. You have to be careful not to throw the bike around too fast. The added rotational movement makes the difference between 2 frames a bit too much for 30 fps to be comfortable. (I play with a high fov on 92" screen so it's more noticeable) The same goes for Wipeout in 1st person mode, needs to be 60.
However MK8, 30 fps is fine. Sure I notice the drop when we race with 4 players, yet after a minute it doesn't register anymore.

As for input lag, that extra 16ms doesn't matter. Racers that focus on 60fps do have a better track record of lower input lag yet that's not always the case. Driveclub is a lot more responsive at 30fps than Project cars at 60fps. NFS Shift 2 is one of the worst I have seen for input lag. Yet going from 30fps on console to 60fps on PC doesn't solve it at all.

One more thing to improve fluid motion perception. Play in a dark room on a screen that's not too bright. You notice frame rate better on a screen with high brightness. Things look more fluent on my projector, calibrated at the image brightness of a cinema screen, than on the living room tv.

Tldr 3 factors affect motion perception
- difference between 2 images, worst in sidescrollers and with mouse look.
- size of screen / distance to screen (VR needs high fps most)
- total brightness

But the point is that there is no input delay in games either as stated before extra frames don't speed the game up, but instead make every motion smoother. The electric impulse that your controller gives to the game itself works at the speed of electricity whether it is 60 or 30 frames per second.

edit: To explain this better here: https://boallen.com/fps-compare.html

I never thought I could tell the difference, but I was playing the Battlefront Beta (60 fps) and then switched back to Destiny (30 fps) and it was super obvious. The game seemed stuttery, but I got used to it after a while.

It's not super important but it IS important. The most important thing in a video game is gameplay, one thing that significantly effects gameplay is frame rate. If a game is choppy and fluctuates at a low frame rate it doesn't matter how good the mechanics/sound/graphics etc are, your experience is ruined. It's the same for online multiplayer network code. Look at BF4, fantastic gameplay, graphics, sound etc, but the play-ability was ruined by poor network code.

Frame rate is one of those things that directly effects your experience.

Having said that, it also depends on the genre. Most RPG/Action/Adventure games run fine at 30FPS (although higher is always better). First Person Shooters and racing games on the other hand benefit significantly from 60FPS vs 30FPS.

In response to the OT, the reason PC gamers bang on about 60FPS/4K gaming etc is because it's one of the few things they CAN bang on about. Don't get me wrong, I am a PC gamer myself, but every 'side' typically finds something trivial to stand behind. With PC gamers its 60FPS/4K, with PS4 owners it's resolution, with XB1 gamers it's XBL, with Wii gamers it's Nintendo exclusives etc. At the end of the day I find it all very boring and pointless and wish developers would sometimes ignore the petty technological complaints of gamers and focus on making good wholesome games.

It's not a must, but the difference between 30 and 60fps is easily noticeable in all games. So it's nice to have.

I was so dissapointed when I played Windwaker HD and found it was only 30fps. It was the reason I bought the game.

Yes, it makes a huge difference. 30fps isn't unplayable. But 60fps just looks and feels so much better. Now that Youtube has 60fps videos, it is very easy to compare the 2 different frame rates. I really do envy people who can't see the difference.

It's not an issue of how long it takes the game to know you pressed a button, it's an issue of how long it has to wait before it can actually show the result of that button press. If a game runs at 30fps, then each frame is on-screen for just over 33ms. Thus, ignoring other factors that can further increase the input delay, it will take 33ms for the result of your button press to be represented on-screen. At 60fps, as well as the general image being smoother, it takes half as long (again, ignoring other factors that increase the delay) for the result of your button pressing to be represented on-screen. If you're pressing several buttons every second, or making constant minor movement adjustments with an analogue stick/mouse, that halved input delay can feel significant.

I don't personally care much for 60fps in most console games, i prefer the benefits targeting 30fps provide, but the advantage of 60fps isn't limited simply to motion detail.