Basically everyone who has played videogames on pc will notice the difference. I easily notice a drop from 360Hz to 240Hz.
I also use 60Hz screens just fine, saying that getting used to 120Hz ruins slower displays is being dramatic. You can readjust to 60Hz again within 5 minutes. But I can still instantly tell which is higher refresh rate, at least up to 360Hz.
Videogames also do the input every loop so there's a big difference there. It must be evaluated with a video only.
We're talking about monitors here, which usually have a mouse cursor on it for input. Of course it would be hard to tell between 60 vs 120Hz screens if you used both to play a 30FPS video.
Lots of games don't do input on every loop. Starcraft 2 has 24 hz input.