Marginally less input lag for one thing: On single events like clicking a mouse button USB polling adds latency up to a full poll interval, i. e. up to 1ms at 1kHz, but only 0.25ms at 4kHz; on average 750 / 2 = 375us faster input. Again, marginal.
Additionally a continuous flow of data as you get with standard mouse surface tracking will appear smoother; an update each 250us as opposed to each 1000us means your cursor travel/game rotation is less jumpy. You could also argue a greater polling rate more accurately reproduces the physical tracking path as lower polling introduces a kind of path correction (see Polling Misnomer: https://www.overclock.net/t/1251156/an-overview-of-mouse-technology
) as per digital sample rate logic. But as opposed to audio or whatever in mousing the reproduction accuracy for "inbetweens" in constant motion is largely irrelevant; you only really care where you end up at and that's the same for all poll rates. Maybe when you are drawing stuff the visually corrective effect could be annoying, but there you can simply move your mouse more slowly to circumvent this effect. Plus, 1kHz is already "path-accurate" enough anyways - at least I don't think artists are maneuvering their hands consciously on a 1ms-scale.
On the flipside, swamping your CPU with 4k+ USB interrupts per second increases CPU load and decreases polling stability/precision (https://www.overclock.net/t/1550666/usb-polling-precision
). Which basically means mouse stutter, noticeable in games primarily.
1kHz is plenty and we don't really need
more. This is just interesting fun and play. Near-instantaneous isochronous communication between host and mouse could still be worthwhile in the future, but for that we'd need a dedicated real-time interface with CPU-independent input processing and mice that internally support sending out their data at rates far beyond the USB standard (i. e. after each individual frame correlation step, which is 6000-12000 times per second?).