So, this is a complicated issue with a lot of ideological elements. First, the modern ethnic group known as "Assyrians" are often said to speak Assyrian, but the language they speak is actually a dialect of Aramaic, whereas ancient Assyrian was a dialect of Akkadian. To confuse matters further, the lingua franca of the Neo-Assyrian Empire and its successor in the Achaemenian Empire was Aramaic (whereas Assyrian Akkadian was used as an official and ceremonial language). It is from this lingua franca, Aramaic, that the modern "Assyrian Neo-Aramaic" descends, along with "Syriac", their liturgical language.

Geographically, Assyria is a region of northern Iraq extending into Turkey and Syria. This was a center of early flourishing of Christianity, and the form of Aramaic known as "Syriac" was adopted as a liturgical language. It is the matter of Christian identity, I would argue, that prevented Aramaic from being completely displaced by Arabic.

So, we should not look at this group as "Assyrians" who survived from the days of Ashurbanipal or whoever whereas the "Babylonians" were wiped out - they are neither identical to nor completely distinct from either group.

As for Median culture, we know very little about it, but it was partly assimilated into Persian culture. There was a dialect continuum of West Iranian languages in this region that was partly displaced by Turkish, and partly developed into what we today call Kurdish.

The desire to identify with particular ancient and prestigious culture is really largely a product of the 19th century, sometimes influenced by hypotheses by various Western orientalists (for this reason, for instance, it is very popular today among Kurds to claim direct descent from the Medes). Such proclamations should really not be taken at face value.