The answer that is common is that China "stopped trying." This however is not where the facts have led me, and I am going to take a few paragraphs debunking:
In the epilogue, Winchester asserts that the consensus opinion of current Sinologists is that "China, basically, stopped trying." That's too facile a summation when one is writing a biography of a man who devoted his entire life to understanding why China failed to capitalize on thousands of years of scientific and technological innovation. Winchester then skips through the main contending theories that attempt to explain China's failure: China's bureaucracy siphoned talent away from a potentially entrepreneurial merchant class, China did not have the spur to competition that Europe's many warring states inflicted on each other, China's totalitarian government quashed initiative.
But Needham himself, writes Winchester, "never fully worked out the answers." (Although he did propose, halfheartedly, a variation on the bureacracy thesis in his essay "General Conclusions and Reflections," a portion of which is online.)
The answer is half there already. One part is that Europe, alone, was situated to reach both Africa and the Americas. Many of the enthnocultures of the new world, which were crucial to the vault to modernity, touched Europe first, and gave them trade goods with China. Tobacco combined with opium for example. China was in the wrong place geologically. Water power, not steam power, drove the first part of modernity. Europe has far more of it in a small space than China does, and must use much less of it for irrigation.
China didn't stop trying, instead, it bet on the wrong things. Europe bet everything on conquest and on metallurgy, only to be rewarded with a continent full of people to conquer, and a series of technologies that came out of this: electricity and the steam engine. Europe does not really vault ahead in most technologies until the early 1800's, I can rattle off a dozen where it will be after that date where China is finally surpassed, but Europe was a global center of empire by 1700, when the outcome of a direct military conflict between China and Europe would have been in doubt, and Europe was still behind in a host of ways.
China also did not have the synthesis of modern mathematics, and this was clear by how Chinese mathematics never reached what is called the calculus.
Lastly, China had a technology that Europe did not have, and it was one that they invested a great deal of effort in: political unity. The Europeans tried over and over again to reach the level of unity that China had, and the level of action that China had, but failed repeatedly, and in a series of wars that history students learn in mind numbing detail. War of Austrian Succession? Of Jenkin's Ear? The Seven Years War, which probably lasted Nine Years?
But while China invested a great deal in the technologies of unity, and was more unified for a great deal longer than its European competitors, it was the wrong choice at that moment. It focused China's energies inward, right at the moment when going out ward would have been most profitable. The crucial century, the 15th century, shows how several innovations in the Eastern Pacific world, canons, print, and the factory, failed to make the same kind of changes in China that they would make elsewhere. Europe, to some extent, locked out of both the unity game and the game of land empire, was forced to turn to the sea, and it was the sea empire that flooded Europe with goods and ideas that would, in turn, give them the confidence to conquer even more.
As Andrew Leonard notes, the age of Europe as overlord is ending, even if you add in the American Century, and a century from now, who knows what the dominant culture will look like globally. If so, perhaps, some future scholar will look back at our moment, and wonder how, with everything in its favor, America "stopped trying."