How religion and science broke up:

I'll focus on Europe for this one.

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The philosophers of Ancient Greece are famous for the many hypotheses they produced on the workings of the world, on the human... So why did they eventually fade away?

Because the greek philosophers were looking for explanations that "make sense". If they had made the jump to doing experiments to test their hypotheses, they would have invented something like the scientific method.
Back in 500 BC.
But they never made the jump to experiments.

Why?

Because it would have violated two subconscious philosophies of theirs:
1. The defining approach to any knowledge back then (and up until the Age of Enlightenment) was that the old ways are best. Anything new and recent is just a poor and corrupted copy of a glorious past full of power, magic and wisdom. Doubting and testing the wisdom of the past was simply considered nonsensical.
2. They had no concept of men being able to use the laws of nature for his own end. The laws of nature were part of the divine realm. The human had no contact to them. Setting up an experiment would have been futile, because your apparatus wouldn't have been able to take advantage of the laws of nature anyways. (All things happen because God wants them to happen.)





It was Ramon Llull, a 13th century mystician, who developed the concept of men being able to control the laws of nature. This was the real treasure hidden in his obscure esoteric teachings.
Ramon Llull inspired a whole avalanche of alchemistic tomes (that were later mistakenly attributed to him).

The Renaissance was the "Age of Rebirth". The Catholic Church had overplayed its hand. It had turned life and worship into ever-expanding lists and doctrines and regulations. People got disappointed. But, isn't the present merely a poor and corrupted copy of the true wisdom of the ancient past?
What if, instead of the bothersome teachings of the Catholic Church, they were able to go back to the original teachings of ancient Christianity?

At this point, a dating-mistake became important. In the 2nd to 3rd century AD, greek Christians had written theological literature on a divine figure, "Hermes Trismegistos". They sought to combine religious knowledge of Christianity, Ancient Greece and Ancient Egypt. But a dating-mistake later lead readers to believe that these books were about an actual mage named "Hermes Trismegistos" who had lived in Ancient Egypt and these were his magical tomes...

Scholars read the books of Hermeticism, believed them to be from many millenia before the coming of Christ... and found hints at Christianity all over the place!
Truly, if someone were trying to go back to the true beginnings of Christianity, this would be the sources to use!

With Hermeticism, magic entered the realm of scholars. Llull had planted the seed that there were laws of nature independent from the divine. And now the "ancient truth" of the hermetic teachings presented an opportunity to access the laws of nature by magical means.
The christian scholar became the mage.

Magic wasn't without controversy, as there was much written about the danger that the powers one might call on might actually be demons. But these warnings were rare: Hermes Trismegistos was a part of the judeo-christian heritage!

What followed was a flurry of research to identify the laws of nature and tap into them. If only one were to combine the religious and magical knowledge in the right fashion, one would become a wielder of magic. This was done in secrecy and obscurity. (This is how "hermetic" became a byword for "sealed-off".)
This is where the idea of a "Renaissance-scholar" comes from: The idea that a scholar were able to understand the whole universe by memorizing the underlying structure of the universe.

Though all those "magical laws of nature" that were proposed "made sense", none of them were able to actually produce "magic". Giordano Bruno was the most prominent of this lot. (The Inquisition wasn't happy that his proposals to return to the true version of Christianity included turning Christianity into a sun-worshiping religion like in Ancient Egypt.)

Magic had failed to produce tangible results.
Where to go next?
At this point, advances in mathematics came to aid, for example by Leibniz. The attitude of the scholars changed once more. A new hunger emerged, a hunger for quantitative solutions rather than qualitative solutions, a hunger for explanations that can be backed up by cold, hard calculations.
Llull's concept of accessible laws of nature, combined with the tradition of the Renaissance-mages to look for them with a method of trial-and-error, combined with a mathematical treatment of the results.
The mage of the Renaissance-era became the scientist of the Enlightenment-era.