The ancient Egyptians would not have encountered neanderthals. Ancient Egypt is generally considered to have started around 3100 BCE while neanderthals seem to have disappeared around 40 000 years ago. So there's a very big gap between the two. Regarding whether they had met neolithic people it more depends on what you mean by that. The neolithic period roughly ends in 4500 BC with the transition to the Chalcolithic. But such transitions are always much more of a gradual process. The Neolithic saw the first agriculture and permanent settlement in the Nile valley and the societies that arose from that were the ones to evolve into what we call ancient Egypt. So in that sense yes, they did meet neolithic people.

Homo sapiens did meet neanderthals, but we have no accounts of them. Writing postdated the neanderthals by many millennia, so no firsthand accounts were transmitted that way. In the absence of writing is oral tradition, but few reach that far back. Aboriginal oral histories possibly reach back 10 000 years (there is some debate on whether that's the case) but even that puts them 30 000 years past the rough time of the disappearance of the neanderthals. Every so often someone will try to claim a certain mythological or legendary creature is actually a disguised memory of the neanderthals, but as far as I'm aware few of those theories are taken seriously nowadays (and in any case, the origin of myths is often pretty much impossible to uncover with any degree of certainty).

But we do know that neanderthals and homo sapiens met. The fossil record for both species overlaps chronologically and geographically. In fact one of the theories as to the extinction of the neanderthals is that the arrival of homo sapiens caused it. Moreover, we know that neanderthals and homo sapiens interbred and most people outside of Africa carry Neanderthal genes, which also shows that there was a fair bit of contact going on.