We've now got about 10 Neanderthal specimens of around 40-50,000 years old that have yielded DNA that is clearly distinct from anyone alive today.
DNA gets everywhere. So when we're dealing with a sample and you find it's got human DNA in it - is that DNA from the fossil, or is it actually DNA from the person who unearthed it?
There's about 1,000 times more mitochondrial DNA than nuclear DNA in our cells, so it's much easier to pick up,