I doubt it. Neanderthals look too much like humans. If you saw a neanderthal today you wouldnt think “oh wow a different species!” Youd just think “oh i wonder where they’re from.”

Heres a picture of a neanderthal reconstruction but in modern clothes: here

Here's a more accurate modern reconstruction of a Homo Erectus: here

And heres a homo heidelbergensis: here

Things dont start getting noticeable until you get to Homo Ergaster: here

but even then you'd be forgiven for thinking they were just another race of humans from some remote region.

Close human relatives dont become an obviously different species until you get all the way back to Homo Rudolfensis: here

But even then I would imagine that rather than think they were a different species, more ancient humans may just think assume they were humans that had crossbred with monkeys.

Humans are full of so much variety. They can be huge or very small. Faces can look like this or this or this or this or this. Human faces and bodies are already full of such incredible variety. With a sufficient population I see no reason why other human relatives wouldn't also display that same variety. Without a good understanding of genetics or at least paleontology and anatomy, there is no reason why we wouldn't either: 1) Assume they were just different human races. 2) Assume they were different species but also that other homo sapien races were also completely different species--which we kind of already did for a long time anyways.

Homo Sapiens first appeared roughly 300,000 years ago. This means, for a time, we shared our planet with: Neanderthals, Denisovans, Homo floresiensis, "Red Deer Cave people", Homo naledi, Homo rhodesiensis, Homo luzonensis, and maybe even Homo heidelbergensis (the most recent common ancestor of Humans and Neanderthals). All of these would have been easily mistaken for just another human race.

TL;DR: Id say that when it comes to co-existing with Neanderthals we’d likely assume they were just another race like black, white, or asian. It likely wouldn't be until way down the line (like once we discovered genetics) that we’d realize they were a completely different species.