Imagine 2 identical gaming setups with 2 players that have the same skill set. In an FPS game, you'd expect each of those players to win 50% of the games.
Now switch one monitor from 120Hz to 240Hz. On average, the player on the 240Hz monitor will see their adversary 4ms earlier than the player on the 120Hz monitor and thus be able to push the mouse button earlier too.
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.
If I’m just watching, I’m not sure I could even tell the difference between 60hz and 144hz.
Further if your network has more than 4ms of jitter then I don't think you can make any concrete claim in either direction.
You can present the game state statistically earlier to the player with the higher refresh rate display.
You can film the screen in slow motion and visually see more fluid motion (and see how it reacts to player input).
Games also use predictive methods and client side hit detection to mitigate most of the effects of network latency in the common cases.
Also, it means each frame is more recent. Also, higher refesh rates reduce the mean variance between the timestamps of rendered frames offered by the GPU that are drawn on the monitor, meaning the visuals are smoother. The smoothness is pleasing to the eye and makes it easier to focus.