Rome was, for more almost a millennium, the most dominant power in the region. For all that Babylon offered in terms of culture, it was far from that.

While Babylon is long-lived, it had not maintained power continuously over that time. It had also been sacked and occupied many times before. From a fairly minor power, Babylon rose under Hammurabi in the early 18th century BCE to a regional power and a cultural innovator, before being sacked by the Hittites in 1595 BCE. In the power vaccuum, a new Kassite dynasty took over for a couple of centuries, yet were fairly powerless and often dominated by their neighbours the Assyrians. Several puppet kings were installed in the Late Bronze Age for example. Eventually the Assryians took control, which is known as the period of the Neo-Assyrian empire around 900-600 BCE. Then there was a resurgence under the Neo-Babylonian empire, but this didn't last more than a century. It changed hands several times after that before the Arabs came through; the Persians under Cyrus, followed by the Macedonians, then the Persians again. It was held for about 900 years in the Persians' second attempt, who allowed Babylon to maintain vestiges of its own culture. When the Arabs took it, most of that culture was destroyed. But it had really fallen long before.

Babylon was a certainly a cultural centre. In terms of law, art, language it was seen, particularly in the 2nd millennium, as a great leader. In fact, this pedigree probably allowed it to continue 'culturally' when its political power was waning. On the many occasions it was sacked, it was kept largely intact for this reason.

So it was not really that Babylon had some great dramatic fall, a cataclysmic event that historians can point to and say, "that was the point when it all went wrong for Babylon". Rather, Babylon had a number of geographic disadvantages preventing it from becoming a great empire like Rome. There were no natural barriers surrounding it, and there were rivals on all sides. It was not really in a position to maintain dominance. So it died a slow death and that's a less exciting way to go for most people, so there is less interest in its demise.

Because by the seventh century Babylon no longer had an empire and had long been a province of the Sassanian Persian Empire (and in other empires before that). Comparable examples might be Antioch and Alexandria, both important provincial cities in the Roman Empire and both former capitals of Hellenistic kingdoms, but their capture paled in significance to the Arab thrusts towards Constantinople in 654 and 668. Babylon was still an important city within the empire, but the fall of the Persian capital Ctesiphon was perhaps more significant. Even then the Persian state fought on for two more decades; though its collapse in Iraq was obviously very dramatic, evidently it did not mark the end of Persian power. Moreover, by then the culture of the region had been transformed - according to the respected historian Hugh Kennedy, Jews formed the majority of the population around Babylon and it was their learning (see the Babylonian Talmud) that was the most renowned thing to come from Babylon in this period. Essentially, Babylon's importance and culture did not stay the same over time, which is to be expected. The political centre had shifted by the seventh century and given our paucity of sources for this period (there are basically no Persian historical sources), historians can just about write a general narrative of the conquest and its impact on more politically/militarily relevant places.

In that sense, the neglect of Babylon is perhaps to be expected. There might be a book out there that describes it in more detail, but there is nothing on this in the up-to-date narratives provided by Robert Hoyland, Hugh Kennedy or Fred Donner. I imagine there might be something in Jewish sources, which are rarely utilised in the study of Arab conquests, but you will have to ask someone else to expand on that. It is also important to note here that our sources for this period are vague and often not helpful, so it is possible that the sources for this event are either not available or not very detailed. The fall of Jerusalem, an important city for Christians everywhere, was for instance only described in two sources two decades later; even then, we know very little about the events leading up to the surrender of the city. This is in contrast to 476, for which we have a lot more sources. It is worth adding here that some historians (and I agree with them) tend to underplay the importance of 476. The Western Roman Empire had been collapsing for a while, so frankly the deposition of a puppet emperor at this point meant very little. Culturally, institutionally and militarily the situation barely changed, so I don't see it as a particularly key moment in Roman history, though maybe this is because it is a topic that has been studied to death...