The fur coat business is a major assumption for which there is no evidence.
One obvious problem, every adult Neanderthal who ever lived had some sort of a little bag of tools which he or she carried around at all times and if any of them had needles, they all would have and the exhaustive searches of tools in the main caves in which Neanderthal remains are found would have turned up lots and lots and lots of them since for any creature which needed clothing at that time, a needle would have been a main tool. One needle in this case simply would not cut it.
http://www.guardian.co.uk/science/2010/dec/05/neanderthals-genome-anthropology
"But which specific traits gave us such an advantage that we were propelled to global glory at the expense of the Neanderthals? In the suite of behaviours that we evolved in Africa 150,000 years ago, what were the characteristics that really made a difference and can therefore be considered as defining human attributes? There are many candidates – complex language and superior memory, for example. However, among many scientists there appears to be consensus that imagination and opportunism were critical attributes.
"This meant, says Fagan, that we learned to use local materials – antler, bone and ivory – in ways Neanderthals simply could not imagine. In one case, this resulted in "one of the most revolutionary inventions in history: the eyed needle, fashioned from a sliver of bone or ivory," he adds. While Neanderthals shivered in rags in winter, humans used vegetable fibres and needles – created by using stone awls – to make close-fitting, layered clothing and parkas: the survival of the snuggest, in short.
The author of course is simply assuming the Neanderthals were "shivering in rags" since nobody has ever found a Neanderthal needle or the numerous such which you WOULD find if they'd ever had them. He was unaware of Vendramini's thesis.