In my answer I'd like to both take a stab at your question, and point out the problems and benefits of other books recommended so far.

First, there are inherent problems with the whole idea of technological "advance." It's really a way to describe only how much a past society is like us, and not a rigorous way of understanding a society's ability to engineer its environment or population for whatever purposes. It therefore contains teleological assumptions, and it is not a category of analysis used by contemporary historians. That said, we can discuss a society's level of economic development in terms of its markets, its ability to project power, and its ability to modify its environments.

One poster recommended William McNeil's The Rise of the West, but I would discourage this unless you're reading it as part of the longer historiography of world history. McNeill offered some pretty potent critiques of his own work twenty-five years after its publication (here]. He noted that his view of history as a diffusion of technological ability through contact was oversimplified, and was very much a reflection of the United States' global position in the late 1950s and early 1960s. Indeed, he makes clear that his chapter on Eurasia from 1000 to 1500 was written from a "naively western viewpoint," and that he therefore missed "the ultimate disturber in world balances" in that period, "an efflorescence of Chinese civilization that raised China's culture, wealth, and power to a new level, far outstripping all the rest of the world for a period of four to five centuries." So, while the book is a classic of world history scholarship, it is not a good recommendation for this question, being out of date by the author's own admission.

Ken Pomeranz's The Great Divergence, on the other hand, is a good recommendation, though it's important to note that he's not especially concerned about abstract (and oversimplified, teleological) notions of technological "advancement," but about economic development. And, he finds that China had freer markets for the factors of production (land, labor, capital), as well as most likely higher standards of living as measured by caloric intake and luxury consumption, until at least 1800--well after 1500. If you want to take that as a proxy for technological capability, that's fine, though I would urge caution in that regard. As Giovanni Federico pointed out in his recent global economic history of agriculture, Feeding the World, while technological development can explain a great deal of the increased productivity of agriculture in the past 200 years, it never presents a linear model of development or a set of changes independent from institutional and environmental conditions. Though he is discussing a different topic, the general point about technology and economic development is a useful one to bear in mind always.

If we're interested in a society's, an empire's, or a state's means to project power, then it is useful to note that Europeans did one thing better than anyone else in the world: build fighting ships. It's been a while since I read these pieces, but the argument is detailed in Geoffrey Parker's The Military Revolution and usefully critiqued in William Thompson's 1999 piece here(pdf). The bottom line is that Europeans did produce better ships and shipboard weapons than anyone else in the world, but that this was about their only area of technological advantage. This allowed Europeans to win sea battles, and at times to take and hold coastal fortresses against Old World opponents in the centuries after 1500, but beyond that their superiority was nil, at least until the 18th century. Overall, keep in mind that European empires, although they grew rapidly after 1500 and could project maritime power, were more or less unable to penetrate the major Asian empires until rather late in the 18th century or even into the 19th century.

Overall, then, with the acknowledged problems of the idea of technological advance, i would argue that there's really no reason to see Europeans as more capable than anyone else until well after 1500. I cannot comment on the comparisons between Middle Eastern and Far Eastern societies, so I'll just leave that.

Also, David Landes's book, recommended elsewhere, is absolute trash, literally one of the worst books I've ever read. Don't bother with it.

It's hard to exactly quantify something like technological advance, but in my opinion the technological development of China, India and Europe was roughly equal around 1500. In measures of industrialization China and India would generally be somewhat ahead of Europe at that time.

Kenneth Pomeranz The Great Divergence: China, Europe, and the Making of the Modern World Economy.

Current historiography, more specifically those historians active in the field of "world history", more or less agree that "Europe" only started off "being more advanced" than other nations starting around 1500. This event is called "The Great Divergence" or "The Rise of The West". So start out by reading "The Rise of the West: A History of the Human Community" by William McNeill and "The Great Divergence: China, Europe, and the Making of the Modern World " by Kenneth Pommeranz. Also read Jared Diamond's work "Guns, Germs and Steel" that talks about why some nations "advanced" quicker than others.

If you want to write a really good essay, I really suggest to also look for academic articles that criticize or nuances the findings of the authors I just mentioned. There's a lot to say about this subject.

Anyway, to come back to your question, historians currently agree that "the West" and "the Rest" were more or less equally "advanced" around the year 1500. So that kind of answers your question. Of course this varies from innovation to innovation and there was a lot of geographical variety even within Europe itself.

In 1500, the East was more "technologically advanced" than the West. (I know that this is opposite of what you started out with but below is my reasons.)

As agentdcf points out, in the 1500s the thing the Europeans were better than the people in the East was warfare. As proof they managed to conquer and hold onto strategic locations despite overwhelming odds. Portuguese Goa in India comes to mind as a prime example. So it is possible to argue that the West was superior in warfare from 1500 to 1900.

However, in terms of products that people paid for with money, there was almost nothing made in Europe that the people in the East wanted to buy from 1500 to 1800.

Of the things the East made, cotton cloth in India, silk cloth in Middle East and China, porcelain in China and Korea, the West could only make inferior copies and thus Europeans bought these goods in massive quantities. And pay for it with silver. In contrast, Asians did not buy European goods. This resulted in a massive negative balance of trade.

Luckily, Spain and Portugal hit upon silver and gold mines in the Americas from 1500 to 1800. There was a big silver rush and a gold rush in their colonies. So that's where the money came from. Silver in particular was important.

Of that silver, A G Frank in ReOrient (1998, p.149) estimates that 25-33% of total world production of silver flowed to China because China exported so much stuff (Chinese silk and porcelain). P. Parthasarathi in Why Europe Grew Rich and Asia Did Not (2011, p. 49) estimates that 20% of total world production of silver flowed to India because India exported so much cotton cloth. So all that silver mined in Potosi (Viceroyalty of Peru) and in Mexico (Viceroyalty of New Spain), went to China and India via Europe.

The trading history of the West from 1500 to 1800 is the story of preventing this outflow of silver (because they believed in mercantilism). The import substitution of the goods that the East made, and also getting the spices that couldn't be grown anywhere else but Southeast Asia, succeeded after centuries of hard work.

Eventually the West made these things in better qualities and more of them and that solved the negative balance of trade and also transformed the world. (Ironically in the 21st century, the Chinese are now back as the main producers of consumer goods.)

In the case of silk: France began to make silk cloth in large quantities in the 16th century at Lyon and eventually made high quality stuff that obviated Eastern imports. French technology finally came up to par with Eastern technology.

C H Mau, London School of Economics paper http://www.lse.ac.uk/economicHistory/seminars/Epstein%20Memorial%20Conference/PAPER-Mau.pdf

(The Italians began making their own silk cloth earlier but their quality never reached French heights and always competed with Middle Eastern imports.)

In the case of porcelain: Europeans always made earthenware, but nothing like the hard porcelain of chinaware. Starting in the 1500s the Dutch, the French, the Germans, and the English all tried to copy it. Example: Dutch Delftware http://www.vannieantiquairs.com/antiques/detail/1211/

vs Ming porcelain http://www.metmuseum.org/toah/works-of-art/37.191.1

The Germans in Meissen were the first to finally figure out how to make real porcelain and that was in 1710. It took almost 200 years after direct contact between Europe and China was established at Macau in 1535 to get to technological parity.

Meissen homepage http://www.meissen.com/en/world-meissen-couture/maison-meissen-couture/300-years-heritage

In the case of cotton: England began to make cotton cloth and that started the industrial revolution. This is such a famous story that I won't bother with too much detail. But the important point to note is that English cotton cloth only began to out compete Indian cotton cloth in the nineteenth century. That's 300 years after direct seaborne trade between Europe and Asia.

West Africa was a place where people got to choose which type of cloth to buy: Indian or English? The English traders who tried to sell English cloth found that West Africans preferred Indian cloths until the nineteenth century. Between 1799 and 1808, British merchants sold 3.2 million pounds of Indian cloth to the Africans and only 2.9 million pounds of English cloth to the Africans. Remember this is the British merchants selling both types of cloth (who obviously wanted to push English cloth). In the entire 18th century, a total of 11.5 million pounds of Indian cotton cloth was sold, while only 7.9 million pounds of English cotton cloth was sold. (Parthasarathi, table from p.25)

So I think in terms of technology in daily life the East was more advanced technologically until the 18th century, and possibly even later. The West, however, did not give up. They worked hard to replace Eastern imports and that eventually led to the industrial revolution which then changed the world.

The real lead in commercial technology came sometime in the late 18th-early 19th century. Before that, there was no technological superiority of the West except in the realm of warfare.

Final point: being better at war does not mean that you are technologically advanced automatically. The Mongols defeated almost everyone they came into contact with, but that doesn't mean that they were technologically advanced. Winning in war could just mean that you are desperate / ornery.

Western superiority in weapons technology is evident by the late 1700s but I think this comes more from the endless wars the Western countries fought.

In the 1600s, the West could not make headway against determined Asian opposition. So I don't see evidence of clear superiority in the 17th century. For example, the Japanese kicked the Spanish and Portuguese out of Japan in the 1630s. The Ming pirate admiral Koxinga kicked the Dutch out from Taiwan in the 1660s. And until 1683 the Ottomans defeated the Hungarians, the Austrians, the Serbs, the Poles, etc and expanded into Europe. (The 1700s is a different story with much better results for European powers, particularly in India. By the 1800s it's abundantly clear that the West is superior militarily, Napoleon's invasion of Egypt comes to my mind for evidence of this. The last 200 years was the height of Western power so we tend to project it all the way back and I think that's a mistake.)

I'm only familiar with one particular technology in this area, but I know from reading Joseph Needham's "Science and Civilization in China" that it was only in the 17th century that sources started to agree that European cannons were noticeably superior to the Chinese variants.

Europeans seem to have developed breech-loading variants first, but the Chinese copied the with some success. By the 1600s, Jesuits in China started noting that Chinese metallurgy was inferior (when it came to firearms,) and that their guns generally didn't perform as well.