Great question. It is indeed often useful to think of Mesopotamia as one civilization, though one that exhibits a tremendous amount of geographical and chronological variation. As I wrote in Why is Sumer so much less iconic than Egypt?,

Few would dispute that Egypt resonates with the public today in a way that Mesopotamia doesn't. Here I mean Mesopotamia from the Early Dynastic right down to the Seleucid period, as it is most appropriately viewed as one civilization with several cultural innovations and evolutionary branches. The Sumerians did not simply vanish with the rise of Babylon, and Mesopotamian scholars continued copying and writing commentaries on Sumerian texts nearly to the end of cuneiform writing. The site of Nippur is a good example, continuously occupied from the 6th millennium BCE to around 800 CE. Although it's best known for its Old Babylonian school texts in Akkadian and Sumerian, Nippur also yielded a large Kassite archive from the Late Bronze Age (mostly unpublished, alas!), and the Murašû archive of ~750 tablets is one of our best sources of documentation on the Achaemenid period. The region of Sumer was still thriving long after Sumerian fell out of use as a spoken language.

Mesopotamia (Greek, lit. "[the land] between the rivers") is the general name given to the region between and around the Tigris and Euphrates rivers. Historians usually refer to two distinct parts of Mesopotamia, upper Mesopotamia (Assyria) and lower Mesopotamia (Babylonia). Babylonia is in turn divided into northern Babylonia (Akkad) and southern Babylonia (Sumer). "Ancient Near East" is the broadest term of all, incorporating not only Mesopotamia but also Anatolia, the Levant, and Iran.

The selection of a term is determined by context. One usually speaks of "Mesopotamian religion," for example, as Babylonian and Assyrian religious practices are so intertwined that it is nearly impossible to tease out separate cultural strands. By the late Middle Assyrian period (ca. 1350-1150 BCE), the Assyrian kings were drawing heavily on Babylonian royal ideology, including rituals and ceremonies, as the weight of tradition lent them legitimacy. Assyria was still a rising power, and the Great Kings of the ancient Near East - such as the king of Ḫatti, who wrote dismissively to the king of Assyria - were slow to accept Assyria as a peer.

You keep speaking about the defeat of Wašašatta and the conquest of the land of Ḫurri. You conquered by force of arms. You conquered by.... So you’ve become a “Great King,” have you? But why do you continue to speak about “brotherhood” and about coming to Mt. Ammana? What is it, (this) “brotherhood”? And what is it, (this) “coming to Mt. Ammana”? For what reason should I call you my “brother”? Who calls another his “brother”? Do people who are not on familiar terms with each other call each other “brother”? Why then should I call you “brother”? Were you and I born of the same mother? As my grandfather and my father did not call the King of Assyria “brother,” you should not keep writing to me (about) “coming” and “Great Kingship.” It displeases me.

The distribution of our sources is even more problematic. Since we have far more sources from the Neo-Assyrian period than from the preceding period in Babylonia (the Middle Babylonian or Kassite period), for example, it is difficult to tell which religious practices were borrowed from Babylonia and which are Assyrian innovations.

In other cases, however, it is important to distinguish between Babylonia and Assyria. Women had different legal rights in Babylonia and Assyria, for example. On a larger scale, Babylonia and Assyria were locked in a battle for control of Mesopotamia from the 14th century BCE onward, a struggle already evident in the Amarna letters (ca. 1350-1330 BCE). In EA 9, the irritated Burnaburiaš II of Babylonia writes to Amenhotep III of Egypt to inform him to send away the messengers from the Assyrian king.

Now, as for the Assyrian, my vassal, it was not I who sent him to you. Why on their own initiative have they come to your country? If you love me, they will conduct no business whatsoever. Send them off to me empty handed.

It's perhaps more helpful to compare Mesopotamia to a region like Mesoamerica than Egypt, which was quite unusual in developing a large centralized state rather early on. There are many cultural elements common to the different Mesoamerican societies - calendars, ball courts, stepped pyramids, the ritual/ceremonial consumption of cacao beverages, and so on - but that does not preclude important differences between the Maya, Aztecs, Zapotecs, Mixtecs, etc. In much the same way, the Babylonians and Assyrians shared numerous cultural elements, so much so that we can speak of an "Assyro-Babylonian" tradition, but Assyria and Babylonia always maintained independent identities, and each had unique cultural characteristics.