I pretty much experience it the way cookybiscuit explained.
I don't see a huge difference, though I do see a difference, but its subtle. However, the responsiveness is huge. I get simulator sickness, which causes me to get nauseated at lower FPS. At 30 it is awful, at 60, it isn't so bad, and I can game about 30-60 mins in 1st person, but if I get to 80+ FPS on a 120hz monitor, I no longer get nauseated.
Another benefit is v-sync does not hurt performance nearly as much as on a 60hz monitor.
But as cookybiscuit mentioned, the best part is 3D Vision. 3D vision brings a game to life. I'll play a game in DX9 with medium settings in 3D over DX11 maxed out any day of the week. There is no comparison, but I have to take breaks after 30-60 mins. due to my simulator sickness.