I can immediately tell just from moving the mouse a little. I wouldn't say it's needed either though.

When I throw my iPhone into low power mode and it drops to a 60FPS cap, it is immediately noticeable.

Many people are more sensitive than you are. I can easily tell the difference between 60 and 120 on both my phone and my desktop.

Though response times also matter a lot for ghosting and such.

Yeah, I'm a gamer and I definitely notice the difference between 60 hz and 120+ hz.

But for desktop productivity? I don't feel I gain anything from it. 60 hz is fine.

60Hz is really noticeable when you’ve been using 120Hz even for a few minutes. 120Hz feels a lot less tiring and work in a terminal, editor and websites is just a lot smoother.

It's immediately noticeable when scrolling in a browser, dragging stuff around or just moving the mouse. If you haven't seen it in person, go to an Apple store and do a quick comparison between the Macbook Pro (120hz) and the Air (60hz), or iPad vs iPad Pro. They're always next to each other.