A while ago I was reading up on
Starblade, one of the first commercial polygon-based games. What really struck me about the game was just how smooth it was compared to its spiritual successor,
Starfox (granted the above links are YouTube videos that don't accurately reflect framerates, but the differences are still quite noticeable).
It's an extreme case, but one that nicely demonstrates the importance of rendering speeds.

- Despite having animations that consisted of only 2-3 frames, many classic games ran at 60fps. This granularity helped to smove out movement, including Mario's beloved jump.
Of course no one ever complains about games being too smooth, but the debate of 30fps vs. 60fps continues to pop up. What's more, the 60fps side keeps losing ground, often to the argument that humans can't really detect more than 30 frames in a single second.
And that is completely untrue.
It's an inherently flawed statement as humans are not digital machines. The human brain is always on, always receiving input. Light hits our eyes as a wave, and the information it carries is almost instantly transmitted to the Visual Cortex. The brain analyzes this data, focusing on changes brought on by motion and fluctuations in intensity. Displacement is interpolated via motion blur and identical input is discarded to avoid extraneous processing.
The "decoded" image is further analyzed by various parts of the brain, but the overall process -- as complex as it is -- is quite fast and versatile. Much faster than 30fps. Faster than 60fps, in fact.
So where does the myth of 30fps come from? Well, film and TV for the most part, but the framerates of those media are not analogous to those of videogames. Others have written extensively about the topic, so I won't go too deep into it. What I'd like to talk about, though, is why high framerates are important to games.

- The Unreal Tournament series has been known for letting its players set very high FPS benchmarks.
As a preface, different titles obviously have different requirements, and some suffer more from a low FPS than others. Also, the reasons for Insomniac's decision to move away from their 60fps standard were completely understandable, if a little painful to accept.
With that said, here's why I think high framerates are important:
1). Granularity
The framerate of a game is usually directly tied to the processing of its logic. As a result, action games that run at 30fps cannot have the same granularity of movement as those that run at 60fps. This might not matter much for turn-based strategy titles, but it makes an awful lot of shmups technically impossible to do at lower framerates.
2). Input Lag
Games are inherently an interactive medium, and as such the response times for input need to be virtually instant. On the hardware side this is rarely an issue, but a stuttering framerate can reduce the response times and greatly detract from the overall experience (especially in "twitch" titles).
3). Consistency
30fps isn't bad, but what most people fail to realize is that it's often the "ceiling" measurement, i.e., the best case scenario. Unlike TV and film, games are dynamic, and the processing required to render any given scene can fluctuate quite significantly. As a result, 30fps games actually tend to run at a rate of 20-30fps. These sort of inconsistencies can be very difficult to avoid, but they're much less noticeable with higher benchmarks.

- Motion blur at its finest.
4). Motion Blur
Motion blur is the biggest reason TV and film get away with smaller framerates. The phenomenon of motion blur relies on the human brain's ability to stitch together a series of blurred images into a single, smooth animation. Until fairly recently, games had absolutely no motion blurring, and even these days it doesn't have quite the same effect. The reason for this is that post-process blurring is not always accurate, and in many cases purposely exaggerated to create a distinctive visual effect.
To properly accommodate for all these factors, a high framerate is a must. And when it's there, it creates a certain synchronization between the player and the game; a smooth flow that more developers should strive to achieve.
Personally, I prefer to play my games on Consoles because I perceive them to be more stable (Frame Rate). Unfortunately, most games on the 360 doesn't look very good in 1080p... I guess i'll have to wait for the next Xbox or PS4.
Quite what I was thinking when the cloud gamming thing came to buzz, it would be nice to just interpolate grafics with 1024-color frame-optimized palletes, updating by grid through different frames... games with less collored palettes, such the grey-brown style of Killzone 2, could use even less colors or channels. Also possible would be just repeat frames that dont have huve frames, and have different Frame Rates and interpolation resolutions for different parts of the screen and different frames. Plus interpolate frames with channels, like frame one is red, frame two is green, frame three is blue... Many gimmicks exploring what the eye can easily ignore in benefit of what it can't.
Though I don't think we need more graphics power than we already have, such services would make MAGs and episodic content very possible.
To have a game running in 30FPS you would have to have film/TV grade renderings (with the simulation of the longer exposure time that comes with 30/24 FPS cameras). Then and only then you could get away with 30FPS, otherwise, you just have a set of stuttering images, like stop motion animations.
Another point I came across, is the notion you should render 60FPS even when the "world state" changes at a lower frequency - that's just wrong, unless you have a genuinely new image to render with new information in it it's obviously not worthwhile - the *only* benefit from this method is to "claim" you are achieving 60FPS ...
over 50/60 FPS is only possible with PCs these days as there are no televisions that can accept a 100/120Hz signal. Even with PCs in the post-CRT era it's hard to come by LCD displays that will do that. Seeing 738FPS measured by the application doesn't mean you're actually experiencing that, just that the computer is doing that much computations ...