I guess that the scientific breakthroughs of DNA analysis are much swifter than the derailments of old ideas.
Apparently, the scientific community is wrestling with the same question.
I had trouble copying this link.
https://www.scientificamerican.com/article/our-neandertal-brethren/

and their offspring are fertile. Wolf-dogs, coydogs, and coywolves are real things, and living among us. The coywolves, especially, are thriving. Here's an article:

https://www.smithsonianmag.com/smart-news/coywolves-are-taking-over-eastern-north-america-180957141/

all the same species?

Humans have created this abstract concept of species, and it is a generally useful idea, but the details of what separates similar animals into two separate species are less precise than many people realize.

Homo sapiens has only existed for about 300,000 years and our branching off from Neanderthals and Denisovans hominoids is recent enough that we could still breed and produce viable offspring. Whether we can even call ourselves separate species may have more to do with human pride than with scientific fact.

We had a common ancestor 300,000 to 700,000 years ago. So we are (were) more or less cousins.

We have learned an enormous amount about Neanderthals in recent years. I highly recommend Kindred: Neanderthal Life, Love, Death, and Art by Rebecca Wrag Sykes. She points out that most of what we know about them comes from bones and stones. Almost nothing else that they had or used or made has survived. But what we've been able to glean from bones and stones, as well as some DNA testing, is remarkable.

but usually the first generation males are sterile. The examples are horse-donkey producing mules, offspring between lions and tigers.

We did share a vast amount of DNA with them, I think around 95%. Better facial reconstructions have shown they might be stared at on a subway, but they really weren't all that different from us. The lifestyles at the time were quite similar and likely technology was shared along with DNA.

Bottom line is that they were human, they were people, and we didn't always pick fights with them.

So would it be safe to say there were enough differences to classify them as separate (human) species, but ones that were more genetically similar than a tiger and a lion or a donkey and a horse?

which is how we get things like designer wild/domestic cat hybrids. The fertility rate isn't 100% for females, which is why we have such a low percentage of Neandertal DNA in modern humans outside Africa. The farther apart the species are, the lower the female fertility rate.

We were very close to Neandertals, about 95%. Fertility among female hybrids was likely common, while males shot blanks. Some studies have suggested an early disappearance of the Neandertal Y chromosome fairly soon after contact, but that hasn't been conclusively proven. Studies have also suggested Neandertal populations had declined to a point inbreeding was beoming a problem by the time we left Africa, but that hasn't been conclusively proven. All we know for sure is that part of that species lives on in us, that we didn't simply see them and slaughter them like people thought we did 50 years ago.

(I'd love to see a friendly domestic cat with the Pallas cat coat and range of facial expressions, but alas, the divergence was too long ago and they're most likely incompatible)

Are there ever fertile female mules?