This may have been covered before so pardon my ignorance. I've always learned about the shifts in early modern thinking from a western perspective but never from an eastern one. Why was the re-discovery of Aristotle so important to the west when the east had the works all along and never had their own renaissance?
A slightly different answer: the Islamic world went through a long period of scientific and technical development, well past the time when the Golden Age of c. 750-950 was over. They continued to develop, slowly, and gradually passed a lot of that knowledge to Western Europe more or less in one fell swoop along with the translations of ancient works. When those European monks translated works from antiquity, they also translated many Muslim works, and adopted them. Western histories typically downplay the strength of Islamic learning and its pervasiveness in the European Renaissance.
One significant example is Copernicus, who didn't just come up with the heliocentric system all on his own: he borrowed a lot from Muslim astronomers.
Because there was nothing to be "re"born, which is the definition of "renaissance". The Greek and Muslim worlds, followed completely different [economic] development paths than the West Europeans.
When we talk about a "renaissance", we are really only talking about the Frankicized parts of Western Europe, originating in France and Germany, N. Italy, and then spreading outwards.
But the simple fact is, that the "Varangian" zone, or that of Scandinavia, simply did not either go through a "Dark Ages", for them to then experience a "Renaissance".
I'd encourage you to read Robert Bartlett's "The Making of Europe", which explains this outward expansion of "Latin Christendom" (I called it a Francization). But the simple fact is, that unlike in Frankish Europe (Charlesmagne's Empire), other areas did not become economically stagnant. It was rather these areas in the Muslim and Varangian zones, that "pulled" Frankish Europe out of the "Dark Ages", through Italian and Hanseatic intermediaries.
This is an incredibly complex question with an equally complex answer.
Firstly, it should be understood that the Renaissance didn't really happen. There was no discontinuity with the later Middle Ages and the classical works preserved in the East had been in circulation in the West for about 200 years by the time the movement called the Renaissance began.
What happened was that these Italians decided that they preferred the long-dead styles of classical Greece and Rome to their living, contemporary styles of art, literature and architecture and started making imitations. Then they started denigrating their contemporary artwork as "Gothic" and calling the era previous to their own a "Dark Age" because they didn't have the benefit of people telling them how much better the Greeks and Romans were.
As far as science are concerned, the transmission of the works of Aristotle was very important, although the role of the Muslims and Byzantines may be exaggerated -- there's some evidence that Irish monks were copying Aristotle before Arab/Byzantine copies arrived -- but it ultimately has to do with religion.
In the Catholic West, there were endless disputes about the nature of God, Grace and the Sacraments, especially the Eucharist. Catholics believed (and still do believe) that the bread and wine at Mass become the real Body and Blood, Soul and Divinity of Jesus Christ, which presents a problem because to all empirical tests, they don't change at all. Orthodox Christians (in the Byzantine Empire) were content to meditate on the Mystery, but Catholics weren't.
There were a couple of approaches: a symbolic or nominalist approach -- the bread and wine are symbolic of Jesus and because we believe in the symbol, it is so; an approach where faith and reason are opposed and irreconcilable (it was similar to Stephen Jay Gould's Non Overlapping Magesteria); and reason and faith are reconcilable -- we can combine Platonic Realism with Aristotelian empiricism to get a substance-accident distinction.
This is scientifically fruitful because it allows us to modify logical theories with empirical data (cf TH Huxley "The tragedy of science is the slaying of a beautiful hypothesis by an ugly fact") and indulge in speculation about abstract things before they have any purpose. For example, mathematicians developed geometries with more than 3 dimensions as curiosities, but it turns out some have relevance to physics.
See Thomas Cahill's Mysteries of the Middle Ages.
In the Islamic world, however, they tended to subscribe to a theology of occassionalism, according to Christopher Beckwith. Unlike Catholics, the Muslims woudn't accept a God who worked through nature, making scientific speculation potentially heretical.