A pro FPS player might notice that they loose contests peeking around corners more often. Obviously network latency in online games will be a factor as well, but since it likely averages out for both players over time, I would guess you can mostly discount it along with alternating who’s doing the peeking.
I don’t think anyone could look at a scene on a 120hz vs 240hz display and tell the difference, there needs to be some indirect clue.