It depends heavily on the type of game and the type of player. For a purely visual experience that looks nice, 30fps is often adequate if the game isn't super fast-paced. However, for a fast action game where you may need to react and aim very quickly (turning the whole player around in an instant), framerate makes a huge difference.
Among other things, if you slew your aim point (point of view) around very fast, even with a fairly high framerate, the view will jump between different points of view, rather than moving smoothly. This makes it difficult for the eye to track, making gameplay more difficult and contributing to headaches and nausea. (Input lag also contributes to these same problems.)
As for latency, from the time you move the mouse to look around or aim at something, it takes at least the length of one frame before you will see it on screen, and up to two frames - if the input is received just as a frame is starting to be rendered based on previous input, it will have to wait for that one to finish before it can even start rendering the next frame, which reflects that mouse movement. It could be even longer if things go through several stages of a "pipeline" of actions before showing up on screen - there could be an additional constant lag on top of the inverse of the framerate. (This is the input lag the article talked about.)
In addition, as pointed out in the article, framerates are often variable, by a factor of 2 or more. So a game that averages 30fps may hit 15-20fps at times (if not lower, though most commercial console games are tuned to avoid this, and many are tested to ensure they rarely drop below 30fps). At 20fps, each frame takes 1/20th of a second to render, or 50ms. So the delay after taking an action before it shows up on screen could be at least 50-100ms (or more, if there's some pipelining), which is starting to be noticeable when you're flicking the mouse around quickly.
If you are playing online against other players, all this is compounded not just by the network, but by the other computers involved. Many actions, including anything that another player did, will be delayed by your control latency (in this case the time from receiving the network information to displaying it on screen), plus the round-trip network latency between you and the server, plus the delay due to the server framerate (how frequently it updates the global state, which might also be only 20-30 times per second - assuming there is a dedicated game server in between, typical for the Quake and Unreal Tournament series of games for instance), plus about the same delay on the other player's end (from the time they give some input to when it is sent to their network connection) plus their network latency to the server.
So even if you only have a 10ms ping time, if you, your opponent, and the server are all running at only 20fps, the total delay before seeing another player's action could be several hundred ms, or over 10x your reported ping time! In this case most of the latency is due to the computers and related to framerate, and not due to the network at all. If all these framerates were 100fps instead of 20fps, then the latency would drop to something closer to 50ms.
So the article that says that most reviewers and casual players (who are generally trying to appreciate what the game has to offer visually, not to compete intensely against other players online) don't notice framerate much is probably true. But the competitive players out there definitely do notice, and will shed any and all graphical amenities if it helps them boost their framerate (= reduce their latency). To them, a slow framerate is like attaching lead weights to a tennis racket.