Sadly, nobody reacts that fast. The benefit of these displays is that they look very pretty.120Hz for FPS games and other high awareness/quick response dependant games it's a godsent really.
For the sake of argument, let's say you're an average sized adult human-being (anybody past puberty). The length between your brain and your finger tips is going to be somewhere around 1m: you've probably noticed most people you encounter are roughly 2m tall and also that your arm span is 'close enough' to the measure from your head to your toes.
Action potentials travel 50ms/s (+/- 20%) in alpha motor neurons (the nerves that send signals to the nerve fibres that make your body move about) in your arms/wrist (see: http://www.ncbi.nlm.nih.gov/pubmed/1557081)
Thanks to the wonders of the metric system we can see that the time a signal takes to travel from your brain to your fingers is 20ms (1m / 50m/s = 1/50s). A 60hz monitor refreshes at 1/60s or 16ms which is already quicker than a signal can get from your brain to your finger tips. if send a signal on frame A, frame A+1 will be displayed before it arrives at your fingers.
that doesn't take into account the processing time that has to go on your brain in order to respond to a stimulus. The travel time was measured by sticking electrodes on nerves, sending a shock at one end and waiting for it to arrive on the other. You don't instantly respond to stimulus that way: photons from your display hit your eye - they get turned into an image, you decide "i'm going to dodge left, that means press my left index finger, arm go! *signal fired*"
Even people you would think to be very sensitive, like race-car drivers or fighter pilots, tend to have similar reaction times to "normal people"(http://www.jneurosci.org/content/30/50/17063.abstract). I think it would be fair to put top-ranked 'videogame athletes' into a category with those sorts of people (ie: the worlds best starcraft player isn't 5x better than the worlds best military pilots). Navy pilots have a reaction time for noticing a 100% contrast 'spot' on a at 350 cd/m (pretty similar to what a good LCD today would be) screen of around 220ms with the best measure times around 140ms (see: http://www.dtic.mil/dtic/tr/fulltext/u2/a178485.pdf). See a change: press a button is ~220ms, travel time for the signal down your arm is around 20ms so that's around 1/5 of a second just to say "oh look, something is different".
What does all that mean? Even if you had a brain 10x more 'sensitive/quicker' than a fighter pilot, and nerves that worked 10x faster than any human being, a 60hz monitor would still draw 2 frames before you could react. In the real world of normal brains and nerves: it makes no difference whether 10 frames or 20 scroll past before you press your kill button: your crumby ape body doesn't have the kind of performance needed to react to changes any quicker than that.
The real reason to want a 120hz screen is because it looks nice. You might not be able to react any faster but you can notice that motion looks smoother (assuming you have hardware to drive > 60 FPS) and you can appreciate the color saturation, viewing angle, contrast, etc.