But to address the OP more seriously:
I don’t see how Neanderthals could exist with us except as a dependent underclass. How else could they fit into our economy? I can’t imagine a Neanderthal living independently, balancing his checkbook, filling out his tax return.
Seems to me they would be reduced to slaves at worst or wards of the state (and worker drones) at best.
Maybe there would there be Neanderthal preserves where they could live in a “natural” state? But what could they hunt there?
There’d be a whole new genre of bestiality. Also, Geico will have lost large amounts of the neandertal market.
If we’re unable to produce fertile offspring with them it could have an interesting effect on sexual taboos. HS woman and HE man sex would probally stay taboo, but HS man and HE woman wouldn’t. HE would resemble us so much that a bestiality taboo might not develop with them.
It might also explain the affluence of fertility clinics ?
We really don’t know enough about them to know what their capabilities were and how they might develop within civilization. After all, if you took a early homosapien and plopped him down in the middle of New York, he’d probably be just as primitive and unable to cope as a Neanderthal. The real question is whether Neanderthals were, like us, capable of doing things like eventually learning complex language and social structures. Many scholars think this is exactly what separated us out from them in terms of our dominance: we had the natural ability and inclination to work together as societies and communicate while their tribes did not do so well as us and got out competed.
Of course, for all we know, they could have all died out because their culture involved eating some ritual animal that got something like mad cow disease. Who knows?
I think a more interesting question is: what would our opinions on animal rights be if the unbroken continuum of species between us and other apes, and apes and other mammals, had not in fact died out, but were still all alive. Would we be as cavalier about killing and destroying the habitats of apes? Would we have this sort of nutty “a human zygote is sacred, but a gibbon baby brain tastes great!” amorality?
Well, the Judeo-Christian concept of humans being placed above and apart from the animals probably would not have developed. Don’t know what sort of philosophy might have taken its place.
I’m pretty sure that we would have enslaved the neanderthals. They would be some sort og low level working class for us
OG SMASH!!! 
The ancients didn’t seem to feel a need to dehumanize slaves in that fashion. They just considered it a natural result of being on the losing side of a war: the conquered are enslaved at the will of the conqueror, sucks to be them, but it doesn’t mean that they aren’t people.
Yep, it’s a skinner. This is a Neanderthal scraper. 
Er, couldn’t they be a bit furrier than Moderns yet still want something extra to keep out the chill? The period wasn’t called an Ice Age because balmy breezes were wafting through Europe.
I’m currently rereading Baxter’s Evolution, which traces the entire evolutionary path of the primates. In one chapter, he has neandertals enslaved by our ancestors. Granted, it’s fiction- but it seems to be well thought-out.
This is assuming that the neandertals aren’t as intelligent as we are, and that we can’t interbreed. If we didn’t enslave them, they’d end up extinct- they use the same resources we use, and live in the same environments as we do, but they don’t compete as well… so unless we took them in as slaves, or pets, or whatever, we’d outbreed them.
If they were as intelligent, or as adaptable as we are, then, given their physical superiority, WE wouldn’t be here. If we could/can interbreed with them, then they’d just be subsumed into our species. If they aren’t as intelligent, or we couldn’t interbreed with them, then we’d make them extinct.
I find it very hard to believe that two, equally capable intelligent species which live in the same environment, could possibly coexist. Whichever one was better at aggressively exploiting the environment would force the other to either become an exploitable resource, or would drive the other into extinction.
I don’t think that one group of sparcely populated hunter/gatherers can enslave another group of sparcely populated hunter/gatherers. Slavery, as we think of it, is a feature of civilization. It might be possible that some members of one clan are captured by another clan, but it’s hard to imagine an entire clan ending up that way.
And it needn’t be one simply solution. I would expect that if there was kidnapping going on, it happened in all combinations, even if one combination was more prevalent. Sometimes the Neanderthal might have outcompeted the Cro-Magnons, depending on climate and terrain. Keep in mind, too, that there was anywhere between 5,000 and 10,000 years of overlap in Europe when both species are found. That’s a long time in terms of what we’re used to thinking of one group wiping out or outcompeting another group. Whatever happened, it was very gradual-- not in evolutionary timeframes, but in modern timeframes of how we’ve experienced interactions between different groups.