According to the current scientific model, Neanderthals were simply another offshoot of the first African "humans", which is where it is believed we all came from.
I don't mean to say that one ape became sentient and started the human race by banging all the other apes and making a race of intelligent apes, only that one species evolved faster then any other and migrated far and wide.
As these primitive humans spread across the earth, they broke off into different groups, had different cities and settlements and sent even further groups of primitive humans even farther into the unknown. This migration took place over a period of time that's exceedingly longer than all of recorded human history, so different levels of adaptation developed different levels of brain power and different anatomical features.
Additionally, different cultures developed as intelligence increased, and questions came to the surface regarding where we came from and why we're here.
It's rational to state that our ability to observe and philosophize, the retarded nephew of science, created the supernatural.
Anyway, Neanderthals were one of those groups that stopped traveling and settled in one region of the world to evolve differently than their brothers and sisters who split off or kept going along the way.
When someone from another group encountered a neanderthal tens of thousands of years later, they would have seen someone who appeared to be a different race of humanoid creatures, but not different enough to not fuck quite a bit, apparently.
The Neanderthal culture is simply what they did with the input they had and the powers of reason and philosophy that they had evolved.
The white man's 4% neanderthal DNA probably gave us some of their evolved characteristics, but since it was more like two breeds of dogs creating a breed with similarities of the two, white people aren't really genetically different than black people, other than what evolution provided to us.