Why does human cultural achievement mean Neanderthal demise?

Again and again, science news publish articles headlining things such as "Humans were genius sexy amazeballs, Neanderthals died of jealousy", or, "Humans wore shoes, Neanderthal extinction due to Osteoarthritis?"

Alright those are fake, but they aren't that much of an exaggeration.

This could have been the very skill that allowed humans
to bound across the European landscape,
ejecting Neanderthals from their cave...

Such as this article in Scientific American entitled "Oldest Arrowheads Hint at How Modern Humans Overtook Neandertals", or this article from National Geographic Daily News published yesterday, "Failure to Hunt Rabbits Part of Neanderthals' Demise?"

The first article I point to is a good example of why I'm not quite following the logic - it is about early projectiles from South Africa, a continent away from any possible human-Neanderthal interaction/competition.  These projectiles would have been a nice fancy cultural innovation for hunting strategy.  And Neanderthals just couldn't cope with their inferior spears... extinction naturally followed.

Sorry, what exactly did the impact of these tools have?  Underlying these articles is just an implicit assumption that life is a boxing ring, and the fighter with the fancier shorts wins (ie doesn't extinct themselves).

Any cultural detail different between our two species seems to be taken immediately to be a marker of why one is here today and the other isn't.  Because, for some reason I can't quite figure out, humans being better at life eradicates other life around it.

But species go extinct all the time.  Why does this one deserve so much attention?  At least that question is easier to understand - our obsession with the demise of the Neanderthals is fascinating, and that fascination leads to endless "what ifs" and fanciful speculations based on little or no evidence.  Neanderthals are us but not us, close cousins that resemble human groups in some ways, but not in others.  They are fascinating.  They are so similar, but they failed.  And we see our differences, our Nectar Points and jet packs, as the reason we have won... something.  Life I guess.  Although, we don't go on about our triumph over the Australopithecines (Husband, pers. comm.)

It's a fascination with different cultural groups that led 19th century anthropologists to come to so many conclusions as to why the 'savage races' were inferior to their own amazing mustachioed culture.

Modern humans have undoubtedly impacted most (all?) species on the planet alive today.  And we seem to extend this impact right back into the Palaeolithic - I can't say whether or not there is a grain of truth in that, and that we did make an impact on our landscapes, but we do have to look at our own species, small in number, without Nectar points or jet packs back then, and maybe have a bit more humility.