This is a question which doesn't fit very neatly into any sub, but r/askhistorians is as good as any other. We're going to have to jump around a bit to answer this one, but it's a genuinely fascinating journey, and the end conclusion will almost certainly surprise you. I'd also like to say nothing that follows is meant to disparage religion.

The first Neanderthal remains identified as such were found in Germany in 1856 (in the eponymous Neander Valley). It was later realized Neanderthal remains had been discovered at least as early as 1830, but had been misidentified. It's no exaggeration to say this sent more ripples through the scientific community than the religious one, as scientists struggled to fit this strange not-man but not-ape into the evolutionary tree. The difficulty was further compounded by the fact Neanderthals and homo sapiens (us - what I'll call "humans" for simplicity) coexisted and were even (scandalously) sometimes buried together. Did we evolve from them, were they the "missing link"? Why then were they still around after we entered the scene, and how did they remain so morphologically distinct? Were they an entirely new kind of human who no longer existed? Possibly, but what a radical idea! One which challenged the sacrosanct idea of us being totally unique and superior in the whole of existence, a belief strongly held by the religious and scientific alike. Were these strange fossils evidence of some kind of disease among humans, or perhaps a very extreme race of humans? Were they closer to what we'd considered a "person", or were they simply anatomically similar non-human apes?

Religiously - and to be clear I'm referring specifically to Christian Creationism - there was already an answer, and like all answers it was to be found in the Bible. Interestingly the Creationist stance was that Neanderthals were as human as anyone else. They were quite simply an extreme product of the Tower of Babel, that episode where all humankind attempted to build a tower that could reach Heaven and, presumably, God. God, as He was wont to do at the time, didn't take kindly to this hubris and (so the story goes) scattered humans all over the world and gave them different languages, to prevent any future uppity architecture. The Neanderthals were simply a far-flung result of this, the Creationists reasoned. They then very obviously met their end during the Flood of Noah. [EDIT: u/DoctorWhoSeason24 correctly pointed out the Flood preceded the Tower of Babel - my bad!]

Having fit Neanderthals neatly into the Bible narrative, Creationists went about "proving" what they already knew to be true. For example an orthodontist named Cuozzo deduced by studying Neanderthal dentition in x-rays that Neanderthals grew constantly throughout their lives and lived to be several hundred years old. Therefore their odd appearance was simply that of a human like you or I who never stopped growing and changing over several centuries. Unsurprisingly these "findings" weren't accepted by the larger scientific community.

It didn't take long after the discovery of Neanderthals for science to roll out yet another "not-us" humanoid: homo erectus in 1891, followed by ever increasing types of humanoids such as Australopithecus and homo florensis (the meter tall humanoids often called "Hobbits"). Creationists shuffled these around for the next hundred years as the variety was, surprisingly, more of a challenge than the single aberration of Neanderthal. As more humanoids were discovered which blurred the line between "us" and apes... well it was starting to look dangerously close to proving the "Evolutionists" correct that humans had indeed descended from apes, which was obviously a non-starter for Creationists. Some maintained that like Neanderthals these were other extinct races of humans, cast out during the Babel episode. Others rethought the past position on Neanderthals being humans and believed all these odd fossils to be various types of (non-human) apes. And still others felt the whole thing was likely an "Evolutionist" hoax, created by mix-and-matching human and ape remains.

There was a belief among many Creationists that back in the days of Eden animals had been much superior to what we now have (what rubbish animals are these days, amirite?). Therefore apes around the time of Creation/Eden/Expulsion were far advanced to what we see today and were capable of walking nearly upright and were in general more similar to the pinnacle of Creation, Man. Unfortunately animals had degraded since then, which explained why apes had regressed to what we have now. Ironically, in their own way Creationists had invented reverse-evolution. However the idea of the process working the other way - primitive animals becoming more "advanced" or modern - was a bridge too far. Although Neanderthals had already been welcomed at the human dinnertable, newly discovered humanoid remains had to be sorted into either "human" or "ape". Unfortunately Creationists couldn't agree which were which.

Perhaps all those that walked upright were human? some Creationists postulated, which was an acceptable and simple enough delineation. Others felt superior ancient-apes could also walk fully upright, and instead we should look at the teeth (dubiously claiming all apes have fierce canines, and those without were human). Others felt we should focus on the brow-ridge; all the fossils have a prominent brow-ridge which humans don't have. Unfortunately neither do orangutans.

The Creationists took all this and went in a very unexpected direction: Neanderthal had already been decided to be human, having had the luck to be first in line, as well as having been found in the rather compromising position of being buried with humans. Creationists were sure humans hadn't evolved, therefore all the remaining troublesome fossils had to be fully either human or ape - obviously there could be no intermediate iterations, as after all God had created Man in His image on the 6th Day and wouldn't have mucked about with ape-men protypes. Creationists therefore decided all the other fossils were indeed those of apes, including that of our true ancestor, homo erectus. That problem sorted, they were ironically left with the same conundrum they faced over 150 years ago - how to reconcile Neanderthals and humans?

The answer, they decided, was that Neanderthals were the original people of the Bible (ie, Adam and Eve, Cain and Abel, Noah and Shem, Abraham and Sarah - all Neanderthals) and we humans are modern Neanderthals. This notion reconciled the existence of Neanderthal and the mixed burials, but did unfortunately little to explain why Neanderthals and modern humans look so strikingly different. They reckoned the reason we no longer see Neanderthal morphology among humans is because we now live a mere century or less, whereas (they decided with the help of those like our dear aforementioned orthodontist) ancient Neanderthals lived to be several hundred years old and grew into their rugged Neanderthal looks (remember that from early on?). This conclusion was bolstered by the many Biblical accounts of people living to be well in advance of 100, up to get nearly 1,000 years old (such as Noah and Methusela).

Much like the apes who'd started out walking fully erect and become more ambulatorily impoverished as they got chronologically further from Creation, the human lifespan had, through a similar degradation, grown shorter and shorter as time went on. If we only lived long enough, Creationists postulated, we'd age into Neanderthals.

So, that's the story of the Neanderthal and the apes. We're Neanderthals - or rather, they're us. Amazingly, as science proves through more and more avenues that not only are we not Neanderthal but we're not even the same branch of human, Creationists find more "evidence" to prove we are in fact Neanderthal. And very ironically our true direct ancestors (as proven by DNA etc) they classify as ape. If you think about it, it's incredibly clever in its obfuscation and at the same time they're so incredibly close to the truth. Anyways, stay strong my Neanderthal brothers.

https://answersingenesis.org/human-evolution/neanderthal/those-enigmatic-neanderthals/

https://www.grisda.org/how-do-neanderthals-fit-with-a-biblical-model-1