Military innovation in the West was driven by a very high share of the economy having spent on fighting wars from the 15th to 19th centuries. Throughout this period, the standard of living and population of Europe often declined due to heavy tax burdens, which in England exceeded 50% of average income at times.

The political condition of Europe following the middle ages was one that lend to constant and incessant warfare between major states, the likes of which did not exist in Asia, India, and to a certain extent the Middle East. In most areas of Eurasia since ancient times, diplomacy had often been dominated by a single large empire in a region, which waged wars against much smaller states, and which did not need to dedicate its full economic might to war. In the 15th and 16th centuries, the rise of the Ottomans in the Middle East, the Mughals in India, and the Ming in China re-created this classic condition. The dominant state in each region needed to dedicate only a fraction of its economy to war, and tax rates were much lower than in most European states.

At times, European politics were also dominated by the concerns of a single large state, but by the 15th century, this condition had collapsed. Though it remained the ambition of certain French and Habsburg monarchs throughout the 15th and early 16th centuries to re-establish "universal empire", neither side was able to acquire a commanding dominance over the other. By the mid-17th century, it became clear that Europe would have no long-term dominant power for some time. The early modern period in Europe experienced constantly changing alliances, incessant warfare, and eventually wars not intended to re-establish universal empire, but to ensure it would never be re-established through the principle of "balance of power". The greater frequency of war in Europe and the greater funding dedicated to warfare naturally spurred innovations. Put simply, it is expensive to maintain large and well-drilled armies of musketeers with cannon, and only the warring states of Europe were willing to consistently bear this expense.

Paradoxically, European military superiority predated European economic growth. Between the 3rd millenium BC and the 18th century AD, GDP per capita growth across all countries and regions averaged zero, as technological growth was always outpaced by population growth. The average English farmer in 1600, in spite of England's leaps in military technology, was no better off than the average Chinese or Indian farmer in the same period, or the average Sumerian farmer thousands of years before him.

There are two prevailing arguments as to why the "Great Divergence" - an event where some countries developed and others were left behind - occured. The older argument, originating in the 1980s, is the "institutionalist" argument. This argument, pioneered by Douglas North, contends that the industrial revolution was really a financial revolution: following the usurpation of the English Crown by William of Orange in 1688, gentries who were tired of losing money on loans to the crown created an oversight body over state debt called the Bank of England. The Bank, through preventing sovereign default, greatly stabilized the English banking sector and allowed capital to be "pooled" for the first time. Before the creation of the Bank of England, banking worldwide had always been a risky activity, as banks had a lifespan that rarely exceeded three decades. The institutionalist argument is backed up by one of the most prevalent models in economics, the endogenous growth theory, which holds that savings = investment, and that investment is the key to long-term growth. Savings only equate to investment if savings are deposited in banks, so it would make sense that humanity only broke out of the zero-growth trap when banking became reliable.

The second argument, pioneered by Oded Galor in 2005, is called the Unified Growth Theory. This theory holds that the end of the zero-growth trap was a result of three factors: first, England acquired colonies overseas which provided a market for English manufactured goods, but were not allowed to manufacture on a large scale themselves. Second, increases in education led to a lower fertility rate, as statistically more educated people tend to have fewer children. Third, increases in education sped up economic growth through an increase in human capital. Fourth, institutional innovations such as the Bank of England did play a role, but not the central one.

Over the course of the past decade, Galor's arguments have been quantitatively verified. Today, most economic historians view them as the strongest explanation for the divergence. They do lead to interesting conclusions, however. While earlier theories on Europe-Asia/Africa divergence viewed industrialization and the end of the zero-growth trap as an objective good that empowered Europe greatly, Galor's Unified Growth Theory revealed that a driving force behind increases in GDP per capita was a decline in population growth rate. In many ways, the success of European colonial empires in economic development also forecasted their eventual dissolution, as Europe's share of global population declined from 25% in 1900 to 10% in 2000.

Sources:

Vries, Peer. Public Finance in China and Britain in the Long Eighteenth Century.

Galor, Oded. Unified Growth Theory.

Black, Jeremy. A Military Revolution? Military Change and European Society 1550-1800.

North, Douglas and Weingast, Barry. Constitutions and Commitment.

Pomeranz, Kenneth. The Great Divergence.

Ashraf, Quamrul and Galor, Oded. Dynamics and Stagnation in the Malthusian Epoch.

Collins, Jason et al. Economic Growth And Evolution: Parental Preference For Quality And Quantity Of Offspring.

This is a relatively commonly asked question on this subreddit, from various angles, and there's some good previous answers:

• /u/profrhodes on Why did European powers, in particular, start dominating and developing at a rate which other powers of the time (Ottoman Empire, Ming Dynasty in China, Saffavids in Persia and the Moghul Empire in India) could barely keep up with?

• /u/commustar on Why didn't Native American and African civilizations advance at the rate of European and Asian ones?

and a now deleted user on The native americans and several other cultures did not experience the same technological advancements as europe. What has caused this diffrence?

Hope that helps!

So, in addition to the links that Hillsonghoods provided, /u/Swarthmoreburke had a very good answer that discussed European trade with Atlantic Africa in the 1400s-1600s. Unfortunately I can't track down that answer.

So, I'm going to link This answer and This answer because they cover aspects of the missing answer.

/u/Swarthmoreburke's explanation was, essentially, that Europeans regarded African societies as complex, powerful, militarily capable. Port cities on the Atlantic and Indian Ocean coasts were cosmopolitan, not backwaters.

/u/Profrhodes comments here, calling that era a "period of incursions", emphasizing the limited nature of European settlement inroads on the continent.

I think I am summarizing these answers fairly to say we all agree that the forces of industrialization in Europe and the massive expansion of the Trans-Atlantic slave trade in the 1700s were the pivotal trends that changed the Euro-African relationship from a situation of relative parity to a situation of African commercial dependency on the slave trade (which was socially disastrous). In the late 1700s and 1800s, there is simultaneously a forgetting of this past situation of parity, and the creation of an ideology of European civilization contrasted with "barbarism" or "underdevelopment" which European philosophers and scholars characterized as the natural state of the world, not a contingent historical development. Swarthmoreburke hits on this in the second answer I linked.