So, apparently the people who answered you removed all their comments. So I'll try and give you some basic guidelines, but this is pretty far from my expertise:

1. First of all, the level of technology did and still does vary drastically across the world.

An important thing to realise is that technology is not just knowledge, it's also the infrastructure to use that knowledge. People like the steppe nomads wouldn't have much advanced technology, because the climate they lived in and the way of life that climate required did not support the development of the complex chains of trade and industry necessary to create more advanced technology. Not because they'd be too dumb or culturally backward to learn it if necessary. (EDIT: or maybe it's more complicated than that, see /u/siqr below)

Also, while I'll leave the why to actual experts, size matters. Eurasia+North Africa was one continuously connected cultural sphere, and because of its size it was generally technologically ahead of the unconnected Americas.

2. Second, technology is not one behemoth. Different people could excel in different things. When it came to shipbuilding I believe the Europeans were ahead of the rest of the world from around the 16th century. But part of this was simply that only the Europeans were interested in building large complex ocean-going vessels. For example the Japanese build a Western-style ship under Jesuit guidance, and [sailed to Mexico in 1614](http://en.wikipedia.org/wiki/San_Juan_Bautista_(ship)). But never had any interest in trading with and exploring the world like the Europeans did.

3. This also moves to the heart of your question. In general, this question is difficult to answer because for a long time what was one of the centres of technology in the Eurasian sphere, East-Asia, was simply not interested in developing the same technologies Europe did. If you're talking general scientific knowledge, then assuming there weren't any highly developed scientific communities I'm not aware of, Europe began pulling ahead from the start of the Renaissance, and you could probably jot down 1600 as a year when Europe's scientific knowledge is the best. The reason I'm picking it is because around then we have the invention of the microscope and telescope, which are the easiest examples I could think of where Western inventions are reaching new horizons in human knowledge.

4. But if you want a clear and simple answer regarding European military superiority: The moment when Europe gained a truly vast and insurmountable lead over the rest of the world is the Industrial Revolution. It is no coincidence that the Opium Wars, the first time China was humiliated by a Western power, happened in 1839. Before the industrial revolution, even though the Europeans were ahead in certain respects, they were not so far ahead that China could not kick them out of their own country if they pleased.

Afterwards, European powers could do essentially what they want, as it was virtually impossible for an unindustrialised nation to withstand an industrialised one.

Why were the comments deleted?