I can't even imagine the impact they would have had on the evolution of our culture and religions if they'd stuck around. The "winning" religions hinge on humans having dominion over all other animals in clear-cut ways.
Might belief systems like Buddhism or Jainism be more hegemonic if there had been a bridge species to tell us we're not so clearly special. And if they'd stuck around, would we still be having such a nail-biting time threading the needle on this extinction event that's predicated on our dominant cultures seeing nature as "other.
I think I'm using the former as a proxy for the latter. If Neanderthals were more distantly related then I would imagine their minds to be more distant from my own. If less, then less. I imagine that my mind is much closer to that of a chimpanzee than that of an octopus, so I would be less amazed by similar behavior in the former than the latter. This isn't to say that I find no amazement in what little we know of Neanderthal culture, just that it's nowhere near the amazement I would find in seeing a truly alien culture. They're on the same continuum, but not at the same point on that continuum.
Well, genetically they were not much different, close enough for interbreeding.
But their mind ... and culture was probably very alien to us.
My point was actually that the DNA difference probably does not lead to a very different thinking. But different culture does.
It's not so much about them being closely related. Rather it's the idea that a mind so distant from my own, that lived a life so alien from my own drew that. I find that deeply fascinating.
The fact that they are lost to us just makes them seem even more interesting, the fact that we can never really know, we can never really understand why they drew that, why they took the time out of their harsh and unforgiving lives to draw some lines on a wall that one day, a different species would ponder over.