I asked my wife if she wanted to have a Neanderthal baby and without missing a beat she said, "I married one, didn't I?"
I searched PubMed, and couldn't find any evidence that anyone has successfully cultured Neanderthal cells yet. It looks like they are closing in on the Neandethal genome, but they don't have a conclusive sequence just yet. I'm very interested in seeing a wooly mammoth some day, and I think that is far more likely to happen. Aparrently, a group got mammoth nuclei to take in some mouse oocytes in 2009. However, I would think that elephant oocytes would be the optimal choice when it comes to a real attempt to bring them back.We have lots of Neanderthal parts around the lab. We are creating Neanderthal cells.
Aside from getting the nucleus to take in an embryo and getting a healthy fetus to keep through gestation, I would guess that the biggest hurdle is the quality of the DNA. Low quality DNA will sooner keep a cell running than reconstruct a living creature. Even if we know the proper complete sequence, I don't think we yet have the ability to synthesize a replacement for genomic DNA. However, there is an abundance of mammoth tissue out there to pick from.
I just realize I posted this link a few hours after you did. I'm just going to delete mind and share yours.