The number of Hertz is the number of how many images your screen draws in one second. So monitor with a refresh rate of 60Hz will draw 60 images each second on your screen. To most users, this will look fine and enough but actually, if you would place them in front of a 120Hz screen difference will be visible right away.
Benefits of higher refresh rates
In most simple words just looking at a screen that has a higher refresh rate will produce a more natural feel of things, although some people would argue that flickering can not be seen on 60Hz monitors and that should be enough, I would argue that they never watched the movie, played the game or used high refresh rate screen when they can state something like that.
Yes, 60Hz might be enough to look at the web page or read some documents but long exposure to 60Hz can have a tiring effect on the eyes in the long run. 120Hz and above just feel more relaxed and it is far less demanding on the eyes. Watching movies also is much better on a higher refresh screen, with way much less motion blur in fast scenes giving the overall experience a greater natural feel.
Gaming is perhaps an area where a refresh rate of 120Hz and above will show its best. Having a higher refresh rate will be easier on your eyes and can even give you a competitive edge. When for example you have twice more images per second on your screen than your opponent his movement will be more fluid and predictable because of that and overall precision will be much higher when moving the cursor in the game since more frames per second means more precision and smoother movement.
The dark side of high refresh rates
After all the positive things that were said about the advantages of high refresh rates, it is just fair to look at some negatives or in this case only negative one and that is Computer power. As said in the previous section having a 120Hz refresh rate means having 120 frames displayed on the screen per single second. Having even higher monitors like 360Hz means even more frames per second and in order to actually see all of that frames you will need to have enough computer power to back it up. Having for example screen with a refresh rate of 240Hz means nothing if your GPU and rest of the computer are able only to push 150 frames per second, in this case, you will still have 150 frames in one second despite the monitor being able to go to 240.
Generally and with all of my heart I would recommend moving away from 60Hz monitors, yes they are cheaper and more affordable but at the end of the day if you are someone like me who spends most of his daily time in front of the computer you would be grateful to have higher refresh rate screen and save your eyes, and after all, why not enjoy some smoother and better movie and game experience as well as saving your eyesight.