That is almost a too broad question!
"Historical writting" Mesopotamia started around 3100/3000 BC and vanished roughly in 549 when the Persian king Cyrius II the Great (an Achaemenid, the movie 300 include Xerxes a later one) conquered Babylon. We could consider the Achaemenid and their successors the Greek Seleucids as Mesopotamians, but I will not here.
Simply speaking the older civilization on Earth. They are renown for creating the very first writing system using clay tablets and coined shaped letters, the cuneiform.
Sumerian texts are generally speaking extremely short (under 100 lines) and obviously rooted in an oral tradition with a distinctive litany style. Their main god was Enlil, a thunder/weather god. Other important gods was Enki a water god, Inanna the love goddess and Nergal the death/war god. Other than that they got a god for almost everything and a specific concept called "Me" which could be identified as roughly the spirit of the things.
Most noticeable texts are:
- The matter of Arratta, which is a series of texts describing the fight between Ur and Arrata.
- The matter of Gilgamesh, which is a series of texts about the hero Gilgamesh (note this is NOT the Epic of Gilgamesh, hence my use of 'matter')
You can find here on SE some descriptions/analysis of those texts by me.
The pantheon of Sumerian gods will be used in all Mesopotamia, it forms also some of the backbone of the early books in the bible (Abraham is orginally from Ur, around 1800, roughly just before the end of Sumer)
It is extremely difficult to separate Akkadians from Sumerians. Akkadians was a Semitic people but came close to Sumer and exchanged/fought a lot with them. In term of mythological stuff, they tapped heavily in Sumerian customs and practices, and bring two new significant things:
- first named poet in history: Enheduanna, which was a daughter of king Sargon is known to have written a praise of Innana (Ishtar). Globally Mesopotamian does not named any poets, so every piece we got are from totally anonymous writer except this piece (if she actually did wrote that).
- The city fall lamentations: The most valuable is the curse of Akkad which describes the fall of Akkad in the hand of barbarians and thus the end of Akkad, you see there some of the gods removing their protection on Akkad.
Akkad is mostly where we find the war goddess aspect of Innana (when she is more a love goddess for Sumerians). It is known in the Bible as Accad.
Understanding the difference between Sumer and Akkad is understanding that: Sumer was a bunch of city state fighting against each others. Akkad was an empire.
Toward 1800BC a small city at the north of Sumer emerged, Babylon, in the late part of the reign of Hammurabi. Babylonian was another semitic people as Akkadian. They will engulf all Mesopotamia in a very short time signing the end of Sumer.
In term of myths, they will tap in the Sumerian/Akkadian (sumerio-akkadian) pantheon, and built on it, adding a whole bunch of gods before the first Sumerian god (An), and add their own tutelary god, Marduk, the Merodach of the Bible, in the middle of it. Note also that while extending the pantheon they paradoxically dramatically reduced the number of gods!
What they bring is: written tradition, Sumerian texts are short and from an obvious oral tradition. Babylonian texts are incredibly longer and much better as work of litteracy.
Most remarkable texts are:
- The Enuma Elish: This is the story of Marduk becoming the chief god of the Babylonian pantheon and the fight between Marduk and his ancestor the dragon/snake Tiamat
- The Epic of Gilgamesh: the story of Gilgamesh's friendship with Enkidu and his vain quest to faint death.
I gave some comparison between Huwawa episode in Sumer and the one that appears in the Epic of Gilgamesh.
Other than that, Babylonian was not truly monotheists, but tended to worship Marduk over any other god, totally disrupting with the Sumerio-Akkadian traditional polytheism.They was users of idols.
Babylon is highly referenced in the Bible. The tower of Babel (probably a ziggurat) is a pretty damn good example.
Assyria is related to Babylon, the same way Akkad is related to Sumer. Assyria was situated in the north of Babylon. The relationship they had with Babylon is so tight that most of the Babylonian/Sumerian/akkadian text we have come from there, especially Nineveh in the library of king Ashurbanipal. For the rest Assyria and Babylon got a love/hate relationship fighting until Assyria vanished. But that also means, like Akkadians, Assyrians tapped heavily in the Babylonian religion to form their own religion. Funily enough there is no per se any "big text book" from the Assyrian.
The main Assyrian god was Ashur, and it is extremely sure he was highly worshiped. Ashur represent the counterpart of Marduk which was also an Assyrian god, but not a big one.
You find roughly the same pantheon than in Babylon. Assyria also brings also specific gods one of them being the fish/wind god Dagon than is used in the Bible and by Howard Lovecraft as a Big Old One. But that is due to their proximity with Hurrians and hittites.
One other point worth mentioning is Assyrian like Akkadian was a pretty military nation, they did engulf Egypt and most of the Hittite empire. And Babylonian was more like Sumer.
Others Mesopotamians can include the Hurrians, and their successors the Hittites, most well known myths is the cycle of Kumarbi (A kind of Sumerian Enlil), well known for being the base of the Greek theogony.
The Persians who brought Zoroastrism after Mesopotamia is conquered. Note that the Persian starting with Darius will use Assyrian as the linga franca, if that shows anything.
Greeks finally brought their own gods.
One has to see Mesopotamia as a very complex land, with lots of history, one region taking precedence, vanishing, coming back, and so on; that what you hear about Assyrians, and neo-Assyrians, Babylonians and neo-Babylonians, Ur I, Ur II, Ur III. Most of the texts we own are deeply rooted in their Sumerian heritage. A good example is taking a look at the most common version of the Epic of Gilgamesh, coming from the late Assyrian Nineveh which is a transcription of the tradition Babylonian Epic, knowing it exists other versions including older one. Or seeing the Enuma Elish Akkadian roots.