There is no visable difference, but the imput lag (time between mouse click and thing on screen) is reduced.

its possible you could experience smoother feeling input

Lets just make up some numbers

Imagine your screen refreshes at 100hz 10 ms per frame with a perfect sync your input will be delayed by that 10 ms.

But what if you ran 200 fps, 5 ms. Well now your input is only delayed by 5 ms because a frame is being drawn every 5 ms and your GPU will only be holding on to that newest frame created every 5 ms to submit to the monitor

This latency would be addative to any latency from mouse/keyboard to computer as well as your monitor's processing time known as input latency tagged as "Lag" in tftcentral reviews

This example Acer Nitro XV273 X review noted that particular monitor they could only estimate potentially 0.5 ms of input latency but marked it as 0 as the estimate was not an actual measurement and 2 ms of grey to grey response times giving it a total input latency of 2 ms. Whereas the average range one may see goes from 3 to 8 ms.

Also worth noting that processing time of 100 ms on a television isn't unusual and that's why TVs are generally not recommended for gaming use.

Input lag will be decreased by up to 1 frame of what the monitor can display.

Say you have a 60hz monitor and try to render at 120fps, the completed images are sent 0.5 frames earlier(counting with 60fps as a base). You try to render at 300fps, the images arrive at 1/5th of the original 60's 1, so you win the remaining 0.8 frames.

As you can guess, this is most of the time not worth it though, as in that last example the GPU still does 5 times the work just for a marginal increase.

If you have a monitor that has Freesync/Gsync, the ready frames get sent immediately, so no need to render above the monitor's refresh rate.

Assuming the time between frames are constant; there is no advantage. Having the GPU output more frames than the monitor can show, just means the monitor will ignore the extra frames. The GPU will do more work and heat your room up faster. So if your room is chilly it might be an advantage, depending what other kind of heating you are using. Most games have a frame limit, vsync or freesync setting, which in their own ways will limit the frame rate to something appropriate.