There's no REAL evidence to support your claim, of "Neanderthals being around long before humans...(Homo sapiens)". And if they had been, for ex. their bones should have been uncovered in the Americas (rather than limited to specific areas radiating from certain points in the Near East), as dinosaurs and other ancient critters have been found here. Their bones should also have been found in Africa & the Far East (where ~other~ varieties of Hybrid bones have been found), had they been present on Earth "long" prior to Homo sapiens.
In fact there is more evidence to support the theory that Neanderthals, like some other subhuman hominids, were Hybrid crosses between Homo sapiens & various types of lower apes. Blood-group & -type research bears it out (eg gorillas, chimps & rhesus monkey populations all share similar blood chemistries w/ Humans that they DON'T share w/ one another: gorillas share bloodgroup B with Humans; chimps bloodgroup A; Rhesus monkeys Rh- bloodtype; & so on); as does the fact that much evidence has been found, of limited levels of fraternization (including sex) between the subhuman Hominids & Human beings.
I believe (based on what I've learned about History & DNA), that there may have been several reasons for ~some~ ancient Homo sapiens to have bred w/ various lower apes. Bottlenecking of our species during the Ice Age(s) may have been one reason.
Some of them may have viewed it as a means to survive & repopulate, in the face of climate extremes which had suddenly wiped out the vast majority of the less well adapted Humans. (Humans don't only wear clothing for modesty, after all. We need it for warmth, too.)
Another reason may have been Pagan superstitions: such as the forces which caused the ancient Hindus to breed the Brahman cattle which they worshipped, & to cover their religious temple walls with frescos of the so-called "monkey armies", a legendary slave 'race' of which the earlier generations depicted in that historical artwork look very much like ~Neanderthals~.
It's no coincidence that Neanderthal bones are found radiating Westward from that geological point. And since they were agriculturalists (farmers quite interested in breeding stock), it's quite logical that they would have engaged in some clever (though often disasterous) genetic experimentation.
If Neanderthals were a separate species from Homo sapiens (rather than as I suspect, a Hybrid species), you shouldn't find any more evidence of them cohabitating with Human beings (or, of them crudely taking on certain Human customs) than is found between Humans and any other lower ape (such as gorillas, chimps or monkeys).
Instead, they seem to have been semi-domesticated by Homo sapiens. Semi-domesticated, because they were too primitive & wild genetically & intellectually to fully domesticate. They were 'almost' Human, more so than any other primates. Yet gorillas, chimps & monkeys still exist alongside Humans & for the most part when left alone, have adapted very well to living in the wild.
And if Neanderthals were a separate species from Homo sapiens, they would be like the gorillas & live off to themselves. Thus they would have survived as a separate species. Instead, traces of their DNA are found among modern Human beings; so they were Hybrids.