No man ever steps in the same river twice. - Heraclitus
Akkad was a long-lasting culture in the Tigris-Euphrates river system, which included a language of the East Semitic family, as well as an empire for about 180 years ending around 2154 BC.
Assyria was a culture centered on the upper Tigris, which included a language descended from Akkadian, and an empire of its own for about 750 years ending around 609 BC.
One could say this is similiar to the modern nations of France, Spain, and Italy, which are modern nations on the same territory as Rome, and speaking languages descended from Latin. But does that make French people the same as Romans? Of course not. A lot of time passed between the two, and in that time there was no small amount of genetic and cultural change. That's why we call them "French" rather than "Roman". There's a big enough difference that its more useful to think of them as separate people.
That's the situation with the Akkadians; Babylonian and Assyrian are both descendant languages from Akkadian1, but in different places. Are they the same people? No, or we'd still be calling them Akkadian. However, they were both cultural descendants of Akkadians.
1 - The literature seems to like to call them dialects or "variant forms" or even a sprachbund, but one could make the same claim about French and Spanish for a lot of their history