The question of why some stories persist and others vanish is incredibly broad and has a corresponding number of potential reasons. I'll list a few.

Babylonian mythologies were often tied up in the authority of the city state and system of government. The end of the system led to the end of the motivating factor for adherence. Most modern large religions, while broad in scope or application, are not tied in to a specific olitical system (catholic Christianity being a potential exception).

Modern religions often have a core of sacred writings believed to be inspired in some way, unchanging, while in ancient Babylon beliefs could be changed on the whim of a king (moreso than today anyway).

Flexibility of application is potentially a good way to describe many of these differences. The abrahamic religions are the product of nomadic peoples who also used agriculture. The religion then developed built on flexibility that still benefits a broad application of the values today.

The shift from "the god of people x" to "the people of x god". This is a shift that can be seen in biblical history, as the Jews transition to viewing themselves as followers of GOD rather than just followers of the god of the land. This leads to religious centric cultures in a way that was different than previous p riestly classes, and this combination of the priesthood of all believers combined with monotheism gives a different kind of strength to how belief and life intersect.

To be continued

Most likely due to the evangelical nature of those Abrahamic faiths and the military and political campaigns behind them. Religious zealotry can be great for morale, meaning that an army of the faithful would be bound to the cross or crescent of saviors and prophets rather than treasure or a tyrant (though nobody was above a little plunder, and tyrants have always existed in all religions). The efforts to eliminate infidels through murder (Islamic invasions & Great Northern Crusades) or social pressures (taxation in conquered Persia and Byzantium after the conversion of Constantine) also helped quite a bit.

That being said, sects of ancient faiths always manage to survive or come back to life in some way or another. Zoroastrians still live in Iran and India even after the Islamic Conquest had quashed the institutions supporting them centuries ago. Assyrian Christians still live in Iraq and the Levant, regardless of conflicts from the 7th century, CE to today. Neopagans (while very different from their ancient predecessors) find inspiration in the Old Gods in the here and now. It could be fair to say that in a more secular Middle East, the gods of Ancient Babylon, Arabia and Persia could make a minor comeback amongst those interested, just as the All-Father has gained a following again in Scandinavians and their descendants in North America.