It would have been a bit more complicated than that. H. neanderthalensis was a cold-weather adaptation which may not have adapted well to the die-off of the large mammals on which he depended. The part about projectile weapons sounds true in that Neandertals show the kind of extensive broken bone injuries that accord with contact killing of large prey.
To put it all in better perspective, H. sapiens didn't exactly thrive, either. We've gone through genetic bottlenecks at pretty much every chronospecific level. Perhaps some Neanderthals were about to rebound, just too late. Some later forms were very different from the earlier examples, some have even been thought to be Neandertal-sapiens hybrids.
Humans succeed best in larger numbers, with more advanced technology. In more primitive terms, we are chancy.
