Would it be possible, theoretically, to “grow” a Neandertal?
Jurassic Park style.
No
Oh I don’t know … my aunt and uncle seem to have managed it. 
I don’t think we have any Neanderthal DNA available. Scientists are not quite sure but the prevailing theory is that they were a distinct species or subspecies of humans.
If Neadertals are an offshoot of mans evolutionary path could we create one by engineering the DNA of a human foetus or have humans lost too many genes since then?
Neanderthal mitochondrial DNA has been recovered and analyzed. However, this DNA was fragmentary, and only limited sequences were sequenced. The analysis suggests that they were not ancestral to modern humans, and could be regarded as constituting a distinct species. (Though this depends on which definition of “species” one is using.)
I am not sure if any Neanderthal nuclear DNA has been obtained or analyzed. If any was obtained, it certainly would be very fragmentary.
If one had the entire intact nuclear genome of a Neanderthal available, it would be possible in theory, by using techniques that have been used to clone other mammals, to recreate something resembling a Neanderthal (although the offspring would have modern mitochondrial DNA). However, we do not have the intact nuclear genome of a Neanderthal, nor are we ever likely to.
Neanderthals evidently were a parallel lineage to modern humans. We have not "lost genes,’ but some of our genes are different from those of Neanderthals. If we had information on every single way in which our genomes differed - that is, if we had an entire intact Neanderthal nuclear genome and sequenced it - it might in theory be possible recreate one through genetic engineering. However, even it we had the genome, we are a long way from being able to do that in practical terms.
The university’s IRB would be pissed.
So, no dinosaurs either, then? 
In one sense, sure: birds are dinosaurs, so we could just clone them.
In the sense you probably mean, not a chance. DNA from the museum-type dinos would be even more fragmentary than Neanderthal DNA - and that’s assuming we could even extract that.
Along similar lines, some scientists in Australia are trying to revive the extinct species, the Thylacine (often called the Tasmanian Tiger), by cloning DNA from museum samples of the animal.
First find a mosquito trapped in amber…extract the Neatherthal blood - the rest is just engineering. If it turns out to be opossum blood… well at least you tried. Get another mosquito… repeat.
Last I heard, people have managed to isolate roughly 100 base pairs of possible dinosaur DNA from amber-encrusted mosquitos. And, of course, it’s quite likely that different samples come from different species, even if they are dinosaurs.
By comparison, the human genome is roughly 3000 million base pairs in size.
Well, we share 98% of our DNA with chimps, right? So we could assume we share at least that much with Neanderthal.
So really 98% of the work is already done, and we just have to find the missing 2%, right?
Note: genome size has little or no correlation to the complexity or size of the animal in eukaryotes. This is known as the C-value paradox. A complete dino genome could be 20 base pairs long. (Not really, but there isn’t any way of knowing how big it is.)
Yes, I know about the C-value paradox. But there’s no way 100 base pairs would even be a drop in the bucket.
without fully mapped Neandertal DNA, how would they know which 2% was in variance?
It doesn’t work that way unfortunately.
First off that 98% figure is just the figure for base pair substitutions, literally where one nucleotide is replaced by another within the same stretch of genome. However substitutions are only a tiny fraction of genetic change. When you also take into a account changes through base pairs being being inserted or deleted the difference doubles to 5%. So really we are at best 95% similar.
However even that isn’t the end of the story. The genome is far more than simply the arrangement of the base pairs themselves. It also consists of the way the genes themselves are arranged relative to one another and within chromosomes. Exactly the same gens exist within humans and chimpanzees, and yet they are expressed in completely different ways because eof physical arrangement as well as interplay with that 5% difference.
Try looking at it like this. You have two cookbooks with recipes for spit-roast pig. And you know that the two recipes are all for basically the same thing. However the books don’t just list the recipe on a string of sequential pages. One recipe for example starts with the instruction “prepare the basting sauce on page 11” and another with “prepare the basting sauce on page 37”. Now when we look at those two sauces they are 95% the same letter for letter.
You can see however we are not in any way able to reconstruct a hypothetical third book that we know is at least 95% the same as these two without reconstructing the whole thing from scratch. That’s because we can never know by examining the two existing books what lappage the basing sauce recipe was on in the original book. It could have been 11, 37 or some other age entirely. And because reproducing the book requires a perfect replication we have no way of doing this. We could certainly cobble together a new book that produces roast pig that is 95% or more similar to the existing books but that won’t make it a copy of the lost book.
Exactly the same is true with trying to reproduce a genome based on genetic similarity. We can safely say that Neanderthals are over 95% similar to modern humans in simple base-pair arrangement, but even there we can’t know what the differences were. But the bigger problem comes because we have no idea how the genes were arranged or how they interacted. We don’t even know something as simple as how many chromosomes a Neanderthal had.
IOW it’s not just the missing 5% we have to find, we have to find the arrangement for the full 100%.
Just an aside, but in Jasper Fforde’s Thursday Next books, neanderthals are indeed cloned by the Jurassic Park method. They turn out to be intelligent but dyslexic, and incapable of violence against other human beings (which explains why they died out). They form lobby groups for 'thal rights (such as the right to hunt mammoths, who have also been revived). It’s probably not the only example from litterature.
The main reason this might be considered to be feasible at all is because there exists a Thylacine pup that was preserved in alcohol rather than formalin, thus conserving relatively undamaged DNA. (However, some deterioration has probably occurred even in this specimen.)
Even with relatively intact nuclear DNA available, as the link indicates reconstruction of a viable Thylacine is still well beyond the reach of current technology. However, given the rapid advancements in this field in recent years, it’s not unreasonable to imagine it might be possible within a few decades.
Although DNA can be extracted from dried specimens, it is going to be much more damaged than that from an alcohol-preserved specimen. And the older a specimen is, the greater the deterioration is likely to have been.
I was thinking we have access to both human & chimp DNA, so we could determine precisely what 2% was in variance between us. Then we could assume the same with Neanderthal. But as Blake pointed out, we’d be creating something that walked like a duck and quacked like a duck, but there’d be no way to know if it were a duck without an actual duck to compare it to.
I could see us making a Jurassic Park by creating dinosaurs that look and act like we imagine they did, but creating something so human-like as a quasi-Neanderthal would I suppose raise a lot of ethical concerns.