What was God doing during 4.5 billion years, before modern humans arose?
It was actually 6 days of messing with things just to trick us into thinking it was 4.5b years.
I'm blanking on all the details. I think "humans" split off. Neanderthals came along, other versions of humans. We happened to get a smaller jaw. Smaller jaw = more room for a brain.
We're smarter than everything else.
To my knowledge, our success is actually the opposite of what you describe in regards to neandertals.
First off, neandertals and sapiens are cousins of sorts. We evolved concurrently, just adapting to different environments. Neandertals populated Europe, sapiens Africa. With Africa having less seasonality, there wasn't a strong necessity for adaptation. The volatile seasons of Europe provided a more challenging environment for survival. Neandertals developed more effective tools and weaponry, and actually had brains 10% larger than ours.
One of the reasons we succeeded is actually because we were DUMBER (kind of...). We were more reliant on a group mentality and division of labor, forcing us into (the original) social networking. It necessitated the development of language and communication. In addition, the larger group size resulted in us having a greater genetic diversity. Immuno-diversity is one of the most important aspects of long term species survival. The larger group sizes also led to larger population gains to a point in which we outnumbered neandertals by a fairly large margin. The rest is history (giggle giggle).
Again, this is just from what I've read. I'm sure there is always a debate going on in the evolutionary community. As a point of information, it's not just the size that matters, it's what you do with it (huh huh). Because of the necessity of communication, our brains developed more complex neural pathways, not just getting larger.
In fact, in recent history, our brains are actually shrinking. [Insert internet commentary about human stupidity here].
EDIT: We actually have neandertal DNA as well. While obviously relevant to our evolution and current state of being, I find this more interesting because of the distinction of species. Species are defined in a way that they can reproduce fertile offspring when bred with one another. This is why a mules aren't a species. Female mules on very rare occasion can produce offspring, but only if sired by males of either species(donkey/horse), not male mules*. My point being that it seems very odd to have any significant amount of neandertal DNA, although obviously possible with the rare multi-generational case described.
*There are no recorded cases of fertile male mules. There have been more than 60 recorded cases of fertile females since 1527, at least according to wikipedia... This seems astronomically low if you would compare it to neandertal/sapien populations at that period, especially if they were in direct competition with one another. But hey, obviously it happened.
Topic: If we evolved ...