I think the definition of species is going to be problematic for some time to come. It is a concept that has no defining lines, obviously, so the blur from species to species, will always be vague, I expect.
That's why we don't really use "species" all that much any more. It's sort of the "child's version" of biology. It's the dumbed down version to help people who don't have enough education.
Clades are more accurate and better represent what's going on in the real world. But, you can't start out someone's education with clades.
When educating children, we are forced to dumb things down and teach in layers.
In the States, very young children learn about "Thanksgiving" - a holiday in which the friendly Indians and the friendly Puritans got together and shared their food.
Is that real? In only the most basic sense. But a 3rd graders isn't sophisticated enough to go into the politics of cultural contact, or the fact that the settlers were starving to death.
Later, they learn a little more.
Then later still they learn even more.
Layers.
You are starting to grasp the basic concept of species. Time for you to get a look at the next level of education.
There is no such thing as "species". It doesn't exist. It's not real. It's a completely made up term invented by humans to facilitate communication between humans.
Without this made up term and the made up rules that surround it, every conversation would have to describe in detail the animal involved.
"Hey I saw one of those flying things that has feathers that are black on the head and it is about the size of my fist"
vs
"Hey, I saw a chickadee".
Nature doesn't have "species".
Nature has different animals which are more or less related genetically. Nature has gene flow between populations of similarly related animals.
Animals which are not closely related have no gene flow.
Animals which are closely related have some gene flow.
Animals which are very closely related have a lot of gene flow.
So, asking whether or not Neanderthals were Homo Sapiens or Homo Neanderthalensis or Homo Sapiens Neanderthalensis is missing the point.
Here's the reality:
They were a population which was closely related to our population. There was some, but not much, gene flow between the two populations shortly after our population left Africa.
Any discussion about how they should be classified is just a discussion between two humans about which socks go in which drawer. It has no bearing on reality in any way.