Early last month Peter Turchin published a couple of posts on whether or not the "Great Divergence" (sometimes called the 'Rise of the West' or 'the European Miracle') can be approached scientifically. Both posts were excellent, and as is usual at Professor Turchin's place, the comment thread that followed the posts was of exceptional quality.  These discussions focused on how difficult it is to use the scientific method to explain an event--like the Great Divergence--that is singular and unique (as opposed to using the scientific method to explain something that has happened many times, like the rise and fall of civilizations generally).
As I reflected on these discussions it occurred to me how important questions are to both the historian's craft and the scientific method. The answers you get depend a lot on the questions you ask. These posts ask "Why did the great divergence between the West and the rest happen?" I would start somewhere different. Before we can attempt to explain why the West diverged from the rest we must ask: which divergence are we talking about?
Most attempts to explain the European miracle start here:
GDP of Western Europe, 1-2003 AD
Between 1760 and 1820 the world changed. I have previously termed this change the "Growth Revolution." As I wrote at the time:
During this revolution human energy production and consumption, population size, wealth, technological capacity, and knowledge all began to increase at an exponential rate. This constant expansion of human resources is the defining feature of our time. Ours is an exponential age.This revolution happened first in Europe, taking the better part of two centuries to spread to the rest of the globe. This time lag accounts for the diverging fates of each region; in a sense the story of the Growth Revolution is the story of the Great Divergence. As Turchin notes, there are a lot of theories explaining why this happened. Many of these run befoul the same problem mentioned in this post's introduction: they answer the wrong question. They jump so quickly to the why of the Growth Revolution that they never stop to ask what the Growth Revolution really was.
This transition changed the dynamics of civilization dramatically. What were once fights over static material resources became a race to see whose resource base could grow the fastest. This is the defining difference between civilizations modern and ancient – the ways war is waged, wealth is acquired, material needs are met, and societies are organized reflect this distinction. The static civilizations of times past faced limitations and challenges alien to our own. 
Luckily this is not a difficult question to answer. Consider the following:
Global Primary Energy Use, 1-2005 AD
Between 1760 and 1820 Europe's wealth exploded - and so did its energy use. The two trends are intimately related. I have discussed the relationship between energy use and wealth several times before , but to re-cap: while definitions of wealth vary with time, place, and culture, wealth is not a mere social construct. It reflects physical goods and services. These goods and services are produced by people or the machines and animals owned by people. We can measure these things in energetic terms because everything a human, animal, or machine does comes at a specific energy cost. "Wealth" is really just the word we use to name the goods created and services rendered through our energy use.
Energy is a universal that unites the natural and human sciences. Ecologists devote a great deal of their efforts to understanding how energy "flows" through an ecosystem. Adopting the same approach when studying human societies provides valuable insights. When human history is seen in energetic terms the source of the premodern world's snail-pace economic growth is immediately apparent: premodern peoples were limited by the energy they had access to. As long as they were dependent on human and animal power their ability to create new wealth would be permanently frustrated. The exponential explosion of European wealth during the 1800s was the direct result of the explosion in Europe's energy use during the same period.
With this knowledge we can amend the original question. "Why did the West diverge from the rest?" is replaced with the more focused "Why did Western nations have the technical and scientific expertise to pioneer non-animate energy sources and an economic system that allowed these new methods of production to spread across the West?"
This is where things get interesting.
The answer to this question extends far beyond 1820. Searching for the origins of 19th century Europe's unique institutions and technology leads to an interesting conclusion: the West began to diverge from the rest long before the Growth Revolution.
This fact is obscured when we look at the West and the rest by the most standard measurement of wealth and power, Gross Domestic Product:
|Estimates in millions of 1991 USD. Taken from Angus Maddison's Contours of the World Economy, p. 367.|
In terms of total wealth Western Europe did not break ahead of China and India until after 1820. Before then Qing dynasty had more wealth at its disposal than every country in Europe combined. From this view it seems pretty clear that the West did not pull ahead of the rest until after their epochal jump into the Growth Revolution.
But this is not the only way to look at things. A different picture appears when we look at GDP per capita:
|Estimates of GDP per capita in 1991 USD. Taken from Angus Maddison's Contours of the World Economy, p. 367 and revisions made in the January 2013 New Maddison Project Database.|
Things were different along the North Sea. Both Great Britain and the Netherlands had sustained GDP per capita growth for more than 600 years. In Great Britain the growth never stopped. Notably, it was in these two regions the technical advances of the Growth Revolution - first with peat in the Netherlands, then with coal in Great Britain - occurred.
This process began before the European conquest of the Americas, the invention of printing, the creation of modern finance institutions, the Atlantic slave trade, or the Protestant Reformation. None of these can be proper explanations for this "little divergence."
So what is? That is the really interesting question. Rather than focus on why Europe diverged from the rest in 1800 we should be asking why the North Sea diverged from the rest in 1000.
EDIT (19/11/2013): Please see the update to this post- "Another Look at the Rise of the West--With Better Numbers."
 Peter Turchin. "The Rise of the West: Science and Ideology." Social Evolution Forum. 3 June 2013 and Peter Turchin. "Cliodynamics of the Great Divergence (AKA the Rise of the West)." Social Evolution Forum. 7 June 2013.
 T. Greer. "Notes on the Dynamics of Human Civilization." The Scholar's Stage. 4 August 2013.
 See "Notes on the Dynamics of Human Civilization" and T. Greer. "Energy Use and Economic Growth: Some Basic Facts." The Scholar's Stage. 13 March 2013. Those wishing for a more technical over view are advised to consult David Stern "The Role of Energy in Economic Growth." CCEP Working Paper 3.10. October 2010.
 The numbers for China and Italy deserve special comment. The Maddison Project economists revised Maddison's estimates for Italy considerably. However, their estimates do not extend to 1000, so I used Maddison's existing estimate for that slot. If their other revisions are anything to go by, $500 is probably much lower than Northern Italy's true GDP per capita in 1000. Likewise, I am inclined to think that Maddison slightly under-estimates China's GDP per capita increases during 1100-1300. See Mark Elvin. Pattern of the Chinese Past: A Social and Economic Interpretation. (Stanford: Stanford University Press). 1979. p. 113-203 for a good summary of the evidence that informs this inclination.