I've hesitated to comment about the sequencing of Neanderthal DNA 'cause I haven't read the papers. Fortunately John Hawks has made the effort and posted the Neandertal genome FAQ. It should answer all your questions, except why John Hawks calls them "Neandertal" when Science and Nature use "Neanderthal." Personally, I prefer the original "Neanderthal."
If you want more information, Nature has a special webpage devoted to Neanderthal DNA.