I don't think this would be a good idea. If it was cloning a simple animal, I'd be ok with it, but we're talking about human beings here. Neanderthals likely spoke like us. They had funeral rituals, used ochre in ceremonies, used tools, and fashioned clothing. They were different from our type of human but still very much human beings.
I think playing around with human life we have to be very much more careful than when trying to clone something like say, the thylacine.