Hi, Archaeologist here. Akkadian and Sumerian were not mutually intelligible, though they did share a very close relationship for a few millennia.

Akkadian is a semitic language, related to modern Hebrew and Arabic languages. Like those languages, it uses the conjugation of three consonants to form verbs, nouns and adjectives, called consonatal roots. It was originally the language spoken by the semitic peoples of Northern Mesopotamia, including later Assyria and Babylon and the city-state of Akkad.

Sumerian is the language of the earliest city-states in Southern Mesopotamia, the originally named Sumerians. The language has no surviving relatives and even at the time of its use there are no attested relatives which have been written down. It is an Agglutinative language, using a complex system of Ergativity and case markers in Verbal and Nominal chains, which in many cases are still not fully understood. It may have had tonality as well, though this idea has begun to fall out of favour.

Sumerian was the first written language in Mesopotamia, perhaps the entire world, depending on whether you ask an assyriologist or and egyptologist. It was widely used for the recording of trade and production from 3500BCE and later developed into a fully-realised writing system for the recording of real speech. For this reason, it was employed outside of Sumeria, including in Akkadian territory, where it was used for the recording of both Sumerian and of Akkadian. Much of the religious writing of the time took place in Sumerian, as even by this point, the Akkadians lent a gravitas to Sumerian much as we did in Europe to Latin, as the language of the bible.

This link is likely compounded by the presence of Sumerian traders and envoys in Akkadian lands and vice versa, so there would have been some measure of bilingualism in the region, though perhaps only amongst educated classes. Now, with the rise of Sargon (𒊬𒊒𒄀 akkadian: Šarrugi) the first emperor of the Akkadian empire, an usurper who changed his name to literally mean "rightful king", the Akkadians came to rule the Sumerian city-states in the South. This accelerated the spread of bilingualism in the region as Akkadians and Sumerians moved about within the empire. What developed from that time onwards has been called a Sprachbund, an area where the languages, though not necessarily mutually intelligible, display a large amount of borrowed words between them. It is believed that during this time, most people would be able to speak at least a bit of both languages.

One interesting example of this would be the Sumerian word 'Gina', meaning 'reed'. It entered Akkadian as 'Qanû', which entered Greek during the time of Alexander's heirs and 'Kánna', then into Latin as 'Canna', old French 'Cane' to modern English 'Cane'.

Eventually, after the 'Ur III' period (a resurgence of culturally Sumerian rule in the region) came to an end, Sumerian would slowly die out as a spoken language, existing only as latin does today, in religious and mythological texts and used widely in monumental inscriptions. Babylonians, such as Hammurabi, would often use 'Sumerograms', which are cuneiform symbols written in Akkadian texts meant to be read using their Sumerian meanings. This was done in order to lend gravity and a sense of timelessness to the writing.

It is important to note that Akkadian didn't remain Akkadian, but diverged into different languages, Assyrian and Babylonian, the latter being heavily influenced by the Amorites, who took control of the region of Babylon, Hammurabi was an Amorite.

Ashurbarnipal, king of Assyria, once boasted "I, Ashurbarnipal, within the palace, understood the wisdom of Nabu. All the art of writing of every kind. I made myself the master of them all. I read the tablets of Sumer and the dark Akkadian language which is difficult to rightly use; I took my pleasure in reading stones inscribed before the flood."

So to conclude, the languages were not mutually intelligible at all, though they were in close proximity for a very long time and borrowed freely from eachother, before Akkadian totally replaced Sumerian as the lingua franca of the region and then was replaced and diverged itself. Sumerian retained a mysticism throughout the later period, which meant its preservation by priests and scribes long after the last native Sumerian had died, even up to the capture of Babylon by Alexander the Great. It is believed that the last writers of Sumerian put down their styluses at the beginning of the Christian era.

Sources: University courses, Babylon by Paul Kriwaczek, Between the Rivers by the great courses A Descriptive grammar of Sumerian, A. H. Jagersma