Nvidia posted an interesting study that takes a closer look at correlation of frame rates and performance in Counter-Strike: Global Offensive, although it's pretty clear that serious esports players require serious equipment to compete.
What you see on the video above is the visual side of things and how different refresh rates make things look, but it's a good indicator of why smoother performance isn't just a vanity thing.
In their post titled Why Does High FPS Matter For Esports, Nvidia highlighted the common misuse and interchanging of frames per second (FPS) and refresh rates in Hertz (Hz), reminding that the former is your system's output while the latter is just your display's capability.
As such, FPS is a dynamic number that changes in accordance with how much juice you've got under the hood, whereas a display's performance is more on the constant side. The fact that these are not in perfect sync naturally causes the issues of ghosting and tearing, the extent of which grows the lower your GPU performance gets.
There's a reason why low-FPS graphics are called choppy - the fewer frames the eye receives, the harder it is for the eye to see it as movement, and when viewed with enough scrutiny - low-FPS footage is more akin to teleporting.
Due to physical design limitations, displays are never instantaneous, i.e. the previous image takes a certain amount to fade, which is called ghosting. The fewer the FPS - the bigger the ghost, and the same goes for tearing.
Tearing, on the other hand, is caused by desynchronised GPU and display, and as you can see from the example above, it displays different parts of a rendered image at the same time. Again - higher performance makes the effect much less prominent, and V-Sync is meant to address these effects.
Ultimately, all of these affect performance and Nvidia's study showed a correlation between performance and K/D ratio although they also stressed that correlation is not causation. We thought the latter was a nice touch because we'd hate for you to think that becoming an esports god is just about buying new gear.