The Sumerians and the Akkadians lived near each other for quite a while, with the culture of the Sumerian city-states being dominant at first. Eventually the Akkadians recognized a good thing, so adopted writing from the Sumerian cuneiform. The Akkadian king Sargon conquered the Sumerians in about 2300BC, so even though the Sumerian language and writing did hang around until about 1800 BC, Akkadian became more and more important and Sumerian less and less.
The Sumerian and Akkadian languages are not at all related: Akkadian is a Semitic language, and Sumerian doesn’t have any relatives that we know of (in linguistics-speak,it is a “language isolate”). The structure of Akkadian was different enough that the Sumerian cuneiform was a bit cumbersome to use to write the Akkadian language. For the Akkadians to read Sumerian logograms out loud, they needed to choose whether they were going to use the Sumerian word or the Akkadian word. Furthermore, while Sumerian had a number of phonetic symbols, they were for the sounds of Sumerian, not for Akkadian.
(Imagine if English were written with a writing system that was partially logographic, partially phonetic, that the French had invented. For the symbol representing flowing water, would I read out “fleuve” or “river”? If I wanted to write down “cheek”, they best I could probably do would be “sheek”, which would sound like the French chic. That’s sort of like the problem the Akkadians had with Sumerian, but French is much, much more closely related to English than Sumerian is to Akkadian.)
It is then perhaps no surprise that the Akkadians made major changes to the script that the Sumerians used. It was still cuneiform, but remember that cuneiform is a writing technology (i.e. reeds and clay) and not a writing system, in the same way that “pen and paper” is a writing technology and not a writing system.
While Akkadians still used a few logograms, they mostly wrote in syllables — there would be one syllable pronounced “ra”, one pronounced “ku”, etc. To write a word pronounced “bakri”, then, you would write the character for “bak” and then the character for “ri”. Syllabic writing systems (or syllabaries) are probably the most common type of writing system among all writing systems.
Here’s a way to impress your friends! Next time you’re at a party and see some cuneiform*, you can tell people if it is Sumerian, upside-down, or a forgery. If there are glyphs composed of more than just the wedge shapes and the little pointy-boomerang things (called “Winkelhaken“), then it is Sumerian. (Particularly if some glyphs are sort of shaded with lines or hash marks.) By the time the Akkadians came along, they only used wedges and Winkelhakens, and the the wedge shapes were very rarely oriented to have the thick part on the right and the thin edge on the left.
*Yeah, right. Like that’s going to happen.