"But how did the white population, much, much younger than the others, manage to gain better technology, health and government that quick? We sort of went from zero to one hundred in five minutes, while everyone else struggled to get up to sixty. Then, with our technology that must have seemed godly to the poor natives, we took over everything."
The key was being united strongly enough to expand, but be divided enough that you had to do it before your rival did. When the French got to southern Africa, they did so to beat the Portuguese to the loot. The British built heavy industry to out gun the French. The worst phase of religious oppression was over and science was being explored as a new way to understand the world. In Europe the religions, technological and cultural changes all came at one perfect storm.
Compare that to the rest of the world at that time. The Chinese had technology, but were such a huge cultural monolith that there wasn't any reason to reach out and change. The Middle Eastern nations had a more fair culture, but were still religiously monolithic, the Africans had the right social and religious conditions, but no technology.
Today the world has just mixed around a little. Western civilization is now a content cultural monolith, China has the perfect mix of national ambition, technology and lack of religious influence, Africa is working on all of them, and Middle Eastern culture is going through its absolutely shitty phase of kicking the crap out of each other for religious purposes (Islam never really had a power mad phase until the 60's. They'll have it out of their system within the next 100 years).
Basically its just the case that Western civilization excelled at the right time. Also can we drop the stupid idea that the rest of the world was its poor little victim? The myth of the 'peaceful little native' is one of the most stupid hippie claims in history. Life in any country was savage, violent and short well before anyone heard of colonialism.