All of these answer are flawed, because they're trying to deal with Europe as if the answer lay with the Renaissances. Romans and Greeks made many of the advancements that the Chinese were also making. It is not who had more technology, East or West, but what it was used for. Europe has seen empires rise and fall, but there has never been a truly united Europe since the fall of Rome. However, there has always been China in one way, shape or form.
Because of that, we can add in this side note: What we know about how advancement in technology has came about to being improved, is too many ways to list, but history can show us that a main drive to improvement is war. Many of the things we now use is proof of that, such as radar, rockets, jets, and microwave ovens, just to name a few.
Back to my main point. China never really had outward enemies to give them the need to advance their weapons to better aid them in battle. They never really were forced into breaking with their old ways because they were perfect as is. This isn't to suggest the Chinese stopped trying to improve themselves, just that the motivation was limited, by both an oppressive state and a lacking of will to break with old ways.
Europeans have for the most part of nearly a millennium, tried to wipe one another out. They have seen the rise of kingdoms like Britain, France, and Spain which for most part tried to gain more or less domination over the others. This really kicked off during the Renaissance with the old knowledge of the Romans and Greeks being taken back from the East Roman Empire after it lost to the turks and a few key imports such as gunpowder from the East, and a weakening of Catholicism with the rise of Protestantism. Plus, many states in Europe started to encourage innovation by rewarding their men of Science, over burning them at the stake for heresy.
European dark ages were a jump backward, but this also led to it rise, multiple kingdoms in Europe never united again (or never for very long); this led to an increase in wars, where kingdoms needed to one up each other in order to dominate
Renaissances led to breaking old ways of Europe as men of knowledge started to disprove many old myths both religious and secular
Weakening of central authority, aka the church, English civil war(both)
Birth of a group of free thinking people who didn't have to dance around rings of the church and the kings
Printing press (@TED gave very good reasons as to why this is)
Nations of Europe advanced themselves to beat others. I'm saying war gave us everything we have, and is very likely to take everything from us one day.