If they lived among us now and interbred with us and produced fertile offspring, they would be considered part of our species, and for politically correct reasons would be re-classified as Homo sapiens sapiens, just like us and lose the Homo sapiens neandethalensis. But, since there is no evidence that they ever did interbreed with our line and they are indeed anatomically distinctly different than us (bigger muscles, brain organized in different ways, skeletal differences, facial differences, etc., etc.) they are, IMHO, correctly considered a different species.
Further, there is evidence that they were behaviorally different than us since their tools ("weapons") were less sophisticated than the ones our direct ancestors fashioned. And while it is PC to say we "outcompeted them" I believe that we wiped them out, and may very well have cannabilized them for good measure (after all, it was much easier to hunt Neandethals than elk or whatever).
It's never been a particular area of study for me, but in my drive by of the information, I don't think we know much about the level of contact. If there was, it seems there would have been sexual contact at some point, but from what I've read, there's no evidence of that. It's also possible that there was sex, but genetically they couldn't produce offspring, or if they did, the child was like a mule, incapable of reproduction, and no samples of DNA from the relatively small subset of offspring have been found. I've read about monkeys using "tools" such as sticks for weapons and leaves to scoop up water, but never of them deliberately assembling or making tools. The Neanderthal tools we've found tend to be somewhere between the monkeys and early humans, as they apparently shaped rocks for tools and such, but didn't go beyond that.
I've seen several reconstructions of neanderthal faces, and they go from being a human ape hybrid to being a face that cleaned up, would be unusual on the street, but not a "stop and stare" face.
With dinosaurs, there's what they were and what we fantasize them to be. I understand that the bird hip and cow hip dinosaurs are both extinct, but I have always contended that if crocodiles had died out before humans lived they would be referred to colloquially as dinosaurs. Various lizard hipped extinct species are referred to as dinosaurs, despite the fact that they don't fit into the strict "dinosaur" structure.
I see things as a continuum, and categories as being useful, but at some point arbitrary. You eventually have to pick a spot and say, "this is where it quits being one thing and starts being something else."