Even if we have the technology to do it, there are lots of ways for the experiment to go very wrong (without going the Jurassic Park route):
…could they be taught in our schools? Who would rear the first generation? Would human parents find this at all rewarding? Do they have enough impulse control to move freely in human society? How happy would they be with such a limited number of peers? What public health issues would be involved and how would we learn about those issues in advance? What would happen the first time a Neanderthal kills a human child? Carries and transmits a contagious disease? By the way, how much resistance would the Neanderthals have to modern diseases?
What kinds of “human rights” would we issue to them? Would we end up treating them better than lab chimpanzees? Would they be covered by ACA and have emergency room rights?
Unlike the debate over recreating extinct animal species like the dodo or the passenger pigeon, Neanderthals were close relations to modern humans: under most of our ethical and moral systems, they would be people, not animals. Unless we’re so debased that we can countenance restarting the Nazi experiment that we forcefully terminated in 1945, we could not treat neoNeanderthals as anything other than intelligent, self-directing, self-owning beings. By bringing them back from the dead, we’d be taking on the moral requirement to maintain them and sustain them.
We have no way of knowing if a group of neoNeanderthals could peacefully co-exist with humanity, and no way of finding that out without running the experiment. That’s not a decision that can or should be taken by a single person or a group of scientists at a university. This wanders too close to “playing god” of old science fiction stories: those stories rarely turned out well for the non-gods.