With the knowledge I have so far about the Neandertals, I'd compare them to Lions: They lived in small groups, had a lot of infighting, and probably little tolerance for members of other groups. 250 000 years of dominating Europe and, under the right climate conditions, surrounding areas, left them pretty unafraid of any other beings - no shyness at all, similar to whales. When modern man arrived, the Neandertals were superior on a one-on-one level, even able to copy some of the newer weapons (as well as tools and cultural objects). They apparently even used complicated methods to glue their spears, among others. Lions with human intelligence and weapons.
Their small groups, combined with some bad weather, was probably their end: When Homo Sapiens attacked them, the survivors could only flee and struggle for survival without any help from other Neandertals. When Neandertals attacked modern humans, the survivors were often able to assemble large parties to hunt and kill the aggressors.
Still, the difference wasn't that big - several thousand years both groups lived pretty much together. The Neandertal was even able to expand into Homo Sapiens territory whenever the climate was suitable. I suppose a catastrophic way the ice age at the time of the extinction of the Neandertals happened had a lot to do with that.
Still I believe there was at least a little bit of mixing - because there still are people with extended eye brows, small chins, and other features usually associated with Neandertals among us.
Judging by those few specimen, the slight differences in looks (of those mixed beings) could even be ignored by society, except maybe the usual teasing among children in school, and beauty ideals among grown ups leading to some disadvantages (but maybe also advantages, due to "exotic" or "savage" looks). Neandertals in such a "mixed" environment would probably have the same jobs and the same capabilities.
If Neandertals survived as a distinct group, for instance by getting isolated on an isle due to the last ice age, and by developing a mistrust against Homo Sapiens which keeps them independent, they'd probably just be seen as a different race today - similar to the Aborigines, and with a lack of higher organisation.
Their hardiness might make them victim to slave traders and army recruiters. Their low population density due to reliance on hunting also is a disadvantage for them. Maybe they could learn from Homo Sapiens in this regard.
Unluckily, most of the isles in question were pretty void of humans until a few thousand or even hundred years ago - probably because of some big catastrophes in the last few thousand years. Not to mention the last 30 000 years.