If you play any games where timing matters, 10ms can be the difference between doing the thing and not doing the thing.

Rhythm games are a perfect example. The tightest timing window is often 1frame at 60fps, which is 16ms. If you are reacting to a headphone with 10ms latency then you'll be missing over half of the timing window. If you also have a wireless keyboard with 10ms then you will react 10ms late, your input will be received another 10ms late and you will miss the entire window and have to adjust your timing to be a full frame early.

Fighting games also commonly use this 1frame window. It's even worse when we are talking about mouse lag interrupting your hand-eye feedback loop on camera movement. I just tried to play the new Myst on an underpowered laptop with too much frame time with vsync enabled and that was enough to make me unable to navigate a curvy corridor, until I disabled vsync.

Latency is a real problem. To put it in the words of John Carmack

I can send an IP packet to Europe faster than I can send a pixel to the screen. How f’d up is that?

The other issue is that you start compounding latency. If you're playing online with a 50ms ping, that hear > react > input registered cycle is suddenly 70ms instead of the 10ms you were expecting. Every single instance of latency you're adding to the system is taking you another step away from reacting in time.