Re: The Neanderthal must have been really drunk...
One thing is clear, fertile interbreeding is the definition of species so we are the same species as Neanderthals, and we're both human.
I'm completely sympathetic to that view as applied to "human", but it's not a sufficient definition of species (which may need multiple definitions)
Consider the Ruddy Duck (USA) and the White-headed duck (Europe). They have very different plumage and have not interbred since the Atlantic ocean became wide enough to prevent these ducks from crosing it. That's maybe 50M years. So, they are recognised as two different species, because naturally they can't interbreed.
But when humans brought Ruddy ducks to Europe, female White-Headed ducks preferred to breed with Ruddy duck males! The hybrids are fertile. So they were once the same species, and no genetic speciation event occurred over the long separation. And if humans hadn't started on a Ruddy duck eradication programme in Europe, soon they'd have still been two distinguishable groups: one Ruddy, one Hybrid, and the European White-headed duck would be extinct.
Or consider the Herring and Black-Headed gulls. You find both in the UK and they don't interbreed. But travel East, and you'll find that the local herring gulls look slightly diffrent. Keep going, and by the time you arrive in the USA, they look a lot like Black-headed gulls. So one species, that's spread around the globe and is overlapping with itself in the UK. But were a catastrophe to wipe out the gulls everywhere except the UK, then two species? (because they won't naturally interbreed here, the complete opposite situation to the ducks).
Or consider the wallabies of Northern Australia, which are the "victims" of a virus which is causing them to undergo genetic mutation at an extremely high rate. There are about fifty species. We aren't quite sure because they all look much the same and inhabit the same evolutionary niche. (How the wallabies know which they are is an interesting question.) Whether there are fifty species with no overlap, or whether small subsets of them are still inter-fertile with other small subsets to an extent sufficient to link all into one or a lesser number of species at the present time, again, we don't know.