Why did Modern Science develop in Europe

Several things:

1.  Europe wasn't quite as bad off as it was made out.  The preceeding ancient Roman Classicaal civilization, was on par with the Chinese, and it was really with the collapse of the Classical Civilization in the early middle ages that Europe fell well behind the Chinese and Muslims for several centuries..  By the High Middle Ages, say 13th century, it had become comparable in level of advancement.  Horizontal axis windmills, reading glasses, were all first invented in Europe by then, not the other places, and Europe Gothic architecture is as technologically advanced as any by the Chinese or Muslims.  Most of the technical achievements were really rooted in the medieval period.

2.  The Catholic Church was instrumental to Europe's rise.  It created the modern universities which became centers for Europe scientific and intellectual advance to this day.  The Church also preserved and carried the legacy of the ancient Classical Civilization to new areas that had been outside the sphere of the ancient Classical Civilization, places like much of Germany, Scandinavia, Scotland.  And it made Westwrn Europe intellectually a common culture, from Poland and Scotland to the more trradition centers of European civilization like Italy.  By the time the Catholic Church turned against advancement, it was too late, the momentum was too strong.   The Church's condemnation of Galileo did little to stop the advance the advance of Copernicus's ideas, for example.  The religious authorities of both the Islamic world and India pretty much squashed the use of printing for centuries - even after the introduction of printing press in both areas, for around 300 years it remained largely used in in the Islamic world and India.

3.  Europe was composed of a number of large poltical units, large enough to promote engineering and scientific advances, but not large enough to stiffle those advances.  If the Holy Roman Empire had decided to ban printing, the printers could always go to Italy or France.  After Columbus was unsucessfull in getting Portugal's backing for his idea, he could go to Spain where he had more luck And if Spain turned him down, he could have go on to England, etc., until he found someone to back him.  The situation was different in China, when the emperor decided against the sea voyages of exploration, it stifled greatly Chinese naval development.

3. As I kind of noted, there was core sense of common identity that promoted the exchange of ideas and discoveries.  The idea that the Earth revolves around the Sun was proposed by a Pole (Copernicus), championed by an Italian (Galileo), and used by a German (Kepler) to calculate the orbits based on measurements made by a Dane.  An international pan Europe effort.  Da Vinci worked in both Italy and France.  The English court painter Holbein was German by birth, as was one the greatest English music composers, Handel.  That kind of free movement of ideas and people certainly boosted Europe intellectual, scientific, and engineering development. 