Like many things in antiquity, the reasons for Babylon's cultural importance can be summed up as "It's important in the Bible, to the Greeks and to the Persians, and then the British arrived".

Babylon first appears in the Bible in the famous Tower of Babel narrative, where the hubris of men attempting to build a tower tall enough to reach the heavens causes God to scatter them by making them speak different languages. The hubris involved in this tale sets the stage for the later historical account of the so-called Babylonian Captivity. King Nebuchadnezzar of Babylon, one of the great villains of the Hebrew bible, besieged Jerusalem in the 6th century and forced the kingdom of Judah into a tributary status. The end of this period - described as one of great humiliation and suffering - is generally where we stop talking of "Israelites" or "Hebrews" and start talking about "Jews".

So, how did this period end? It ended when King Cyrus of Anshan, in western Iran, conquered Babylon, for which he is much praised in the portion of the book of Isaiah known as "Deutero-Isaiah". This account, which is extremely hagiographic, in my opinion very likely postdates the reign of the third/fourth Great King of the Achaemenid Empire, Darius I, as its rhetoric is very reminiscent of Darius' propaganda. The early history of the Achaemenid Empire is very blurry and heavily distorted by the contrivances Darius invented to justify his rule; the narrative accounts that survive portray Cyrus as an incredibly just, brilliant and tolerant ruler, which actually does not mesh particularly well with more contemporary (but fragmentary) records found in Babylon. This is a very complicated issue which I can expound on at some other point.

Anyway, Babylon became the administrative capital of the Achaemenid Empire, the most powerful empire that had ever existed; it was unsurpassed in extent until the Umayyad Caliphate, and thrived for 200 years, until it was brought down by Alexander. Around the founding of the Achaemenid Empire begins what is known as the Classical Greek period, during which culture and philosophy flourishes in Athens. Though Athens was never conquered by the Persians, there must have been much influx of culture and knowledge from the east. For Greek authors (who generally did not successfully differentiate between the Persians and earlier rulers) Babylon was a source of astrology and mystic wisdom, owing to its great antiquity. The Greek literary figure known as Pseudo-Zoroaster, a heavily distorted version of the Iranian prophet, was in many accounts described as a wizard and/or king of Babylon who had lived thousands of years even before the Trojan war, credited with inventing magic and astrology. Babylon, therefore, plays an important role in the Greek canon.

So Babylon played an important role in the Western mind from these sources, yet -- nobody really knew what Babylonian culture was like, and therefore imagination could run wild. In the 19th century, however, this begins sto change, as the ruins of Babylon began to be excavated, concurrently with the emergence of Egyptology as a discipline (a consequence of Napoleon's invasion of Egypt). While I don't think Babylonian artefacts were ever revered the way Egyptian ones were, Babylon was a treasure-trove of written tablets. Curiously, the aforementioned propaganda of Darius I in the form of the trilingual Behistun inscription, served as a Rosetta stone for the decipherment of Babylonian (Akkadian) writing, since educated guesses could be made about the form of Old Persian words.

This places the conception of Babylon within a complicated context, not only straddling many ancient civilizations of great importance, but also being an important archaeological site, which has complex ideological implications in its own right, given the context of imperialism, the vast amount of artefacts sent to museums in Europe, the fact that Babylonian was a Semitic language (like Hebrew and Arabic) which in turn held racial-biological implications to 19th century thinkers in the West, and so on.

Yeah, I don't really know how to wrap this up, and I wish I could turn it into a more coherent narrative, but I hope this gave some amount of insight into the Western view of Babylon and its prominence. I recently bought a 1000-page corpus of sources about the Achaemenid Empire ("The Persian Empire"); the importance of Babylonian records for understanding it can hardly be overstated!