Just a couple of observations:
1) H. sapiens had the ability to organise large communities from the beginning. So clans &/or tribal units of over 50 - 100 people operating as a single community, which was common place, has a large advantage compared to all the other humans around, such as Neanderthals et al, who operated on a family type level of somewhere between 6 - 12 individuals (give or take).
2) H. sapiens were (are) great innovators where it seems Neanderthals et al weren't. This probably comes from our ability to be able to think abstractly, where our ancient human cousins may not have been able to think at such a high cognative level. Being able to think abstractly also allows for advanced problem solving, not to mention constantly trying to improve what is currently the case; whether it be a tool, a weapon, or more importantly, team work at both the tactical & strategic level. Neanderthals, however, once they had developed something to a certain point then, like H. erectus, no longer pushed the limits of their tools, weapons, & so forth, unlike our immediate ancestors (H. sapiens).
3) Knowledge. Although Neanderthals seemed intelligent enough, they didn't seem to accumulate knowledge & pass it on to the next generation in great volumes. Instead, like H. erectus, only the basics were passed on & not much else. H sapiens it seems, like their current descendants (in other words us), placed great emphasis upon education beyond merely the basics.
4) Taking the three points above, it shouldn't be surprising to note that H. sapiens could rapidly adapt to various environments in a very short period of time. From the deserts of Africa, to the snow covered mountains of Europe, to the hot rainforests of Asia, there you'll find H. sapiens. Our other human cousins, however, seemed to evolve into "specialists" & thus were more or less forced to remain within a certain region. If the environment, however, rapidly changed in their region, they were in trouble. H. sapiens, on the other hand, was (is) a jack-of-all-trades specie, which may lack particular "specialist" skills, but can nevertheless do pretty much everything anywhere. This means H. sapiens actually has a higher chance of survival as we can deal with many environments efficiently when required, whilst the other species of humans can only deal with a few.
Essentially, the way I see it, H.sapiens was merely a refinement. We had a little bit extra than the others where it mattered in the 4 areas above. In the short term, maybe Neanderthal &/or H. erectus may have had particular advantages in one or two skill/ability areas, but in the longer term, H. sapiens ability to organise, innovate, think abstractly, & accumulate knowledge over several centuries & pass it on, plus an ability to be highly adaptive to any given environment, gave them a better chance to survive in the long term in a not so friendly & highly competitive world. And if there's one certainty with evolution, a slight advantage is all that's needed to survive.