Firstly it depends on the game. If it's an RTS or racing sim, they'll probably play at a higher resolution as there's no real downside. When it comes to competitive FPS, it's all about performance.
The difference between 200FPS and 600FPS can be massive even on a 60Hz monitor due to physics engine behaviour. Quake III Arena was the first I know of, where getting 333FPS made you jump higher and every other footstep silent, along with some other bugs. One of the Call of Duty games had something similar where you could jump further at a refresh rate above a certain number. There are a ton of games that inadvertently give an unfair advantage when the frame rate exceeds what the physics engine is capable of.
Over and above that, when you're playing at the level that milliseconds are the difference between winning and losing a duel, the difference between 144Hz (7ms/frame) and 360Hz (2.8ms/frame) can be a winning advantage. Maintaining 360FPS on many games is hard even with high end hardware, which is why a lower resolution is used. That and the fact that there's no such thing as a 4K 360Hz screen.
There's also the 1% and 0.1% lows. If your 1% lows are 15% lower than your average frame rate and 0.1% lows are 40% lower, the differences work out as follows:
Average Frame Rate | 1% Lows | 0.1% Lows |
600FPS (not linked to refresh rate) | 510FPS | 360FPS |
360FPS | 306FPS | 216FPS |
165FPS | 140FPS | 99FPS |
In this example, the 600FPS player's worst experience is equal to the 360FPS player's best. Those drops aren't unrealistic, as there are some game and hardware configurations where 1% lows can be 50% below average and 0.1% lows can be 80% lower than average (there are heavier games where it doesn't matter what hardware you have, at high resolution you'll hit 1% lows of 1FPS - I'm excluding that). That leads to the following:
Average FPS | 1% Lows | 0.1% Lows |
600FPS (not linked to refresh rate) | 300FPS | 120FPS |
360FPS | 180FPS | 72FPS |
165FPS | 82FPS | 33FPS |
When playing at a level where 5ms determines who wins, a 30ms frame-time will have a very negative outcome.
All in all, a more expensive, higher resolution monitor is a disadvantage.