Each jump saves less time
A refresh rate is a time in disguise. Divide 1000 by the rate and you get the milliseconds each image stays up: 16.67 ms at 60 Hz, 8.33 ms at 120 Hz, 4.17 ms at 240 Hz and 2.08 ms at 480 Hz.
That's why the gains shrink. Going from 60 to 120 Hz takes 8.33 ms off every refresh. Doubling again to 240 Hz takes off 4.17 ms, and the jump to 480 Hz takes off just 2.08 ms. Each doubling still halves the time, but you're halving a smaller number every time.
The popular upgrade from 60 to 144 Hz trims 9.72 ms per refresh. The FPS Visualizer puts any two rates side by side.
A refresh rate is a ceiling
A screen can't show more frames than it refreshes. Play at 200 fps on a 144 Hz monitor and you see at most 144 of them, one every 6.94 ms. If your graphics card makes only 90 fps, a fixed screen repeats frames to fill the extra refreshes, and a screen with VRR slows down to match. Either way you see 90 different frames a second. A 240 Hz monitor won't turn a 90 fps game into a 240 fps one.
Unlike a CRT, an LCD doesn't flicker at a low refresh rate. Its pixels hold their state, so the refresh rate mostly decides the highest frame rate it can show.
Video has its own quirk. Movies are shot at 24 fps, and 60 doesn't divide evenly by 24. A 60 Hz screen uses 3:2 pulldown, holding frames for 3 and 2 refreshes in turn, so they stay up for 50 ms and 33.3 ms instead of an even 41.67 ms. That's the slight judder in slow camera pans. A 120 Hz panel holds each film frame for 5 refreshes, which is 41.7 ms every time.
What it costs your cable
Every extra refresh is more data down the cable. Take 4K at 10 bits per color. At 60 Hz it needs 15.7 Gbps uncompressed, at 144 Hz 39.2 Gbps and at 240 Hz 68.6 Gbps. HDMI 2.1 carries 42.6 Gbps of data and DisplayPort 1.4 carries 25.92 Gbps.
So 4K at 144 Hz doesn't fit DisplayPort 1.4 as it is, and the cable has to fall back on Display Stream Compression. HDMI 2.1 fits the same mode without it. The Display Bandwidth Calculator checks any resolution, refresh rate, color depth and cable.
Check what your system is really running
Windows often starts a new monitor at 60 Hz, so a high refresh screen can sit there until you pick its top rate in the display settings. What Hz Is My Monitor reads the rate your browser is getting, so you can check that the change took.