Even if you were more closely related to Neanderthals, that isn't necessarily a bad thing. Neanderthals were, in fact, physiologically superior to modern humans in a number of different areas. Just as Modern humans are superior to Neanderthals in a number of different ways. It is yet to be proven in favor of or against the theory that Homo Neanderthalensis is in fact Homo Sapiens Neanderthalensis, thus allowing modern humans and Neanderthals to interbreed, however if by some chance you are more closely related to them, my guess would be that the fact that you actually exists means that your genes are just as good as everyone else's on average according to the laws of natural selection.
The reason that the the Neanderthals died out wasn't due to an intellectual or physiological weakness. They died out basically as a result of competition for food in a changing environment.
Since the Neanderthals required around 2-3 times the energy needs of a modern day human being, the competition for food scored in favor of modern day humans (or the closest thing at the time- the cromagnons) who did not have the same amount of muscle mass for insulation, since they evolved in a warmer environment.
The Neanderthals also tended to dwell in the valleys in order to conserve energy by being closer to food sources and not having to metabolize at a higher rate to keep themselves warm. Cro-magnons, however, use to reside mainly in the higher mountain ranges, which ended up being a greater advantage, for although they were using more energy, the view from these heights allowed them to survey their territories more effectively.
Probably one of the most significant factors that contributed to the downfall of the Neanderthals was the changing environment. Towards the end of the Neanderthals' reign over Europe, the landscape began to quickly change from a forrest-dominated region to open planes. Since the cromagnons evolved in an area where this was already very much the case, they were already very prepared for this. The spears that the cromagnons possessed were far lighter than those that Neanderthals possessed, making them far better suited to traveling across open savannas. The spears of the Neanderthals were to heavy, large and slow. Since there were so few neanderthals, the likelihood of altering the craftsmanship which had so far sustained them was slim. The cromagnon population was also far more fertile, since there was a greater degree of genetic variability.
Thus, Neanderthals- like almost every other extinct species on earth- did not die out due to inferiority, as is commonly misconceived, but rather due to the random nature of genetic adaptations that pre-conditioned cromagnons to be better suited to a warmer climate. If we were to have another ice-age (or very significantly cold weather), however, without heating, you would find that those who were built like Neanderthals would be most likely to survive the longest. There are some clips on youtube which exhibit this.
If it is proven that Neandethals are a sub-species of the Homo Sapiens species, as Homo Idaltu has already been proven to be, then this debate about superiority amongst Neanderthals and modern humans would almost be the equivalent of a racial debate, and thus is reminiscent of so many historical clashes between races. Many sources argue that this is definitely not a possibility with the availability of the Homo Sapiens Sapiens and Neanderthal genomes available, however there is in fact not enough viable evidence to prove this without a doubt.
It is important, then, to remember that in the case of genetic variability amongst extreemly closely related populations, superiority in most cases is a fruitless argument. For, populations are only superior in their adaptations to their native environments and thus, inferior in their adaptations to new or different environments. The only exception to this would be the chance of a random genetic mutation, which pre-disposes individuals within a population to exceed their peers in their ability to survive.
Hope that this helps.
Cheers,
Fifteendaysofbliss.