This is an interesting debate that is still ongoing. There are three main positions. 1) Neanderthals did not wear clothes, 2) Neanderthals wore simple cape-like clothing and 3) Neanderthals wore complex clothing similar to early modern humans. Even though bone needles are not known from Neanderthal sites, some archaeologists have argued that awls and cutting tools would have been enough to have shaped and connected skins together to make the tailored clothing needed to survive in glacial periods in the Eurasian region the Neanderthals inhabited.
An article published in the Journal of Anthropological Archaeology in 2016 added new data to the discussion. Collard et al compared the animal remains found in Neanderthal and early modern human occupation sites with an ethnographic database of fur used by more recent non-industrial communities for cold-weather clothing in a similar latitude.
The animals weer not identified by species but by family, which meant that extinct species from the Palaeolithic could be included in the comparison. Fur and skin from the deer family seems to have been the most widely used material used for clothing in both Neanderthal and early modern human assemblages, followed by that from the cow family, and then a number of families of smaller animals, namely weasels, rabbits and dogs. There was also the use of bear skin, squirrels, cats and beavers.
There was a significant difference between the occurrence of the weasel, rabbit and dog families at Neanderthal and early modern human sites and the researchers suggest that the latter were more commonly used by early modern humans as fur trimming, confirming complex tailored clothing. The fact that bones from some of these families were found at Neanderthal sites, especially the weasel and dog families which are very infrequently used as food in more recent non-industrial societies, suggests that they did wear clothes. But the very low numbers of these bones found at Neanderthal sites points to them not creating complex cold-weather clothing.