So if you want to take a continuous (analog) signal, digitize it, then convert back to analog, you are fundamentally adding latency. And if you want to do DSP operations on a digital signal, you also generally add some latency. And the higher the sampling rate, the lower the latency you can achieve, because you can use more compact approximations of sinc that are still good enough below 20kHz.
None of this matters, at least in principle, for audio streaming over the Internet or for a stored library — there is a ton of latency, and up to a few ms extra is irrelevant as long as it’s managed correctly when at synchronizing different devices. But for live sound, or for a potentially long chain of DSP effects, I can easily imagine this making a difference, especially at 44.1ksps.
I don’t work in audio or DSP, and I haven’t extensively experimented. And I haven’t run the numbers. But I suspect that a couple passes of DSP effects or digitization at 44.1ksps may become audible to ordinary humans in terms of added latency if there are multiple different speakers with different effects or if A/V sync is carelessly involved.