It's super easy, put your finger on a touchpad and move it fast in circle so that the cursor also moves in circle. As the eye is not that fast, you will see multiple faint mouse cursors images. With 120 Hz there will be twice more cursors than with 60 Hz.
On a perfect display you should see just a faint grey circle.
Another test is moving cursor fast across the white page and tracking it with eyes. On a perfect display it should be perfectly crisp, on my display it blurs and moves in steps.
So basically on a perfect display you can track fast moving things, and when not tracking, they are blurred. On a bad display, things blur when tracking them, and you see several instances otherwise. For example, if you scroll a page with a black box up-down, on a bad display you would see several faint boxes overlayed, and on a perfect display one box with blurred edges.
You could replicate a "perfect display" by analytically implementing motion blurring (which is really just a kind of temporal anti-aliasing) in software. This wouldn't let you track moving objects across the screen without blur, but that's a very niche scenario anyway. Where 120hz really helps you is in slashing total latency from user input to the screen. A 60hz screen adds a max 16.667ms of latency, which is plenty enough to be perceived by the user.