All of this is well covered in the FAQ sections on Civilizations of the Americas and Why is Africa "less developed"?

I believe the question has been asked before in about 1000 different ways.
The answer is very complicated. If you look at the late 15th century, as you suggest, Western Europeans (and their genoese hired sailors) were not more technologically advanced than several other groups, such as the Chinese, Indians, various groups of middle-easterners, japanese, etc etc - but they were experimenting with some advancements in ocean vessels.
The gap started to widen, with fewer and fewer analogs to european tech and power, and by the start of the 19th century only (parts of) China and Japan could boast tech and power that was strong enough to hold off the brits, dutch, and french.
The three elements that were nicely isolated by Tonio Andrade in Lost Colony that set Western Euro's apart were:

1. The broadside ship - lots of guns, ability to tack against the wind - this was key in an indian ocean system in which the europeans most often acted as pirates - one ship could do a lot of damage and could easily chase down trade vessels that could only get away if they had the right wind.
2. Star-shaped forts - this allowed the portuguese and dutch to hang on to strategic land positions long after they ought to have been able. The forts were easy to construct with low walls, but the cross-fire capabilities made them impregnable without a major artillery supported siege.
3. Artillery works - the modular engineering work that was done to support the forts helped hold off much larger armies. In the case of the dutch in taiwan, whom Andrade writes about, it took nine months for a vastly superior (numerically) force to take the Dutch fort, and it was only after the Chinese began building their own artillery works to counter the Dutch positions in the most vulnerable side of the fort.
   But, these tech's weren't enough to handle China at the time, and they wouldn't be enough until the 1840's. The Brits tried several times in the 1830's to cow the Chinese on the Yangze and coastal areas, only to be foiled by fire ships and the like. Then around 1840, the Brits ascended beyond the power of any that came before - mostly due to improvements in firearms and steam power. By 1850 they were unrivaled, and all others would be following on their coattails. The steamship was so superior to anything that came before that the brits instantly dominated formerly intransigent foes (Chinese, Sudanese) By 1880, most western euro states and japan had caught up, and the chinese weren't far behind.
   So, my answer really centers on China, and it is mostly a cautious warning not to underestimate China before the first Opium war.