Neanderthals were a comparible species to modern humans in many ways, and although were physically different in some ways, were thought to be very similar to humans. One of the major technological and developmental differences however is that Neanderthals were not thought to have ever developed or used any projectile weapons.
One of the main reasons that Neanderthals are thought not to have developed projectile weapons is due to the fact that the European hunting environment was a lot different to the African one. While Cro-magnon man had already developed throwing spears and atlatl (commonly known as spear launchers) when hunting some species in Africa, this probably wasn't as necessary in many parts of Europe.
Africa being comprised of savannas and grasslands a lot more, would have meant that chasing down prey was a lot more difficult as they had more space to roam about. It might not always have been possible to land a blow on a prey species in order to would it prior to giving chase. This eventually lead to projectile weapons being a necessity, and the use of throwing spears would have slowly spread across the population.
In ancient Europe, the Neanderthals tended to stay more in a central location rather than roaming from place to place in search of prey. This also must have meant that there were constant food sources, as well as meaning that the prey species must have been easier to catch in some way. This was perhaps due to the fact that there were a lot more forests, boggy conditions and generally slower moving species. It is thought that Neanderthals would wrestle with their prey often leaping onto it before stabbing it with their spears and knives.
Cro-magnon man was also able to spread new technology a lot better than Neanderthal man was, as by the nature of being a nomad, they would have had frequent contact with others of their species. Neanderthals were much more stationary, meaning that even if some small groups had used projectile weapons, this probably wouldn't have spread to the entire population.
Eventually all areas of the world developed some kind of projectile weapon, from the slings and bows of eastern Europe, to the spear launchers of Africa. Even the relatively isolated aboriginal people of Australia developed the boomerang, and yet despite all this the Neanderthals did not develop anything of the sort. However for the most part projectile weapons would have likely emerged as a response to difficult hunting conditions.
However the fact that Neanderthals were technically a separate species to humans means that the same expectation couldn't be made of them. The same way that we couldn't expect all primates to use tools in the same way that orangutans do, there is no reason to think that Neanderthals should have developed projectile weapons merely because modern man did.
The main reasons that projectile weapons have been created through the ages was firstly as a means of hunting and secondly as a weapon of war. However since the Neanderthals didn't need them for hunting in the fairly cold and heavily forested areas of Europe they inhabited, they were never developed. Similarly Neanderthals would never have engaged in any kind of war in any major way, as they existed only in fairly small groups and not tribes or communities.
Although Neanderthals probably threw stones and some spears at their prey, for the most part they seemingly preferred to use longer, thicker thrusting spears. However given that they were a lot stronger than modern man, they probably didn't need to fear getting close to their prey as much as they were better able to withstand punishment.
The fact that the Neanderthals did not develop any projectile weapons is thought by many people to be indicative of their lack of technology. However more likely they lacked the motivation and need to develop them rather than lacking the ability or potential. Many other forms of Neanderthal technology are comparable and in some cases superior to modern humans of the same era, for example their ability to make extremely sharp spears.