There's always more to be said on the topic, but I touched on this in How exactly did Aramaic somehow replace Akkadian and become adopted by much of the Neo-Assyrian Empire?

Aramaic had long been the language of administration in what became the western portion of the Assyrian empire, and it was easier to maintain that administrative infrastructure than to attempt to force everyone to begin speaking and writing in Akkadian.

There are quite a few parallel examples of this phenomenon. Probably the most obvious example that comes to mind is the use of Greek as an administrative language in the eastern portion of the Roman empire. The Romans had no intention of forcing everyone in the eastern Mediterranean to use Latin when administrators in the region had been using Greek for centuries. It's primarily in western Europe, where writing was limited or nonexistent prior to the Roman period, that we see the adoption of Latin (e.g. the Vindolanda and Bloomberg tablets in Britain).

It should be noted that Aramaic never fully replaced Akkadian in the Neo-Assyrian period; rather, the Assyrian empire was a multilingual empire that routinely used both Akkadian and Aramaic, sometimes even on the same documents. Akkadian survived alongside Aramaic down to the end of the Assyrian empire, as exemplified by the very late Akkadian and Aramaic texts found at sites like Tell Sheikh Hamad.

Aramaic continued to flourish after the collapse of the Neo-Assyrian empire as the main administrative language of the Achaemenid Persian empire, and it gradually replaced Akkadian until the latter died out entirely by the 1st century CE.

For further reading, see Languages of Iraq, Ancient and Modern edited by J. Nicholas Postgate.