Armenians and Assyrians were one folk together as Subartians (People of Subartu), a Hurrian type of people. The first speakers of the Hurrian-Urartian language family are considered to be the Kura-Araxes culture.
https://en.wikipedia.org/wiki/List_of_Assyrian_kings#Early_Assyrian_rulersThe Early Assyrian Kings List features the names of "kings who lived in tents," implying that they were a Semi-Nomadic people. The names of these Assyrians kings are most likely written in the Hurrian language. So it would make sense that the Assyrians were originally apart of this Hurrian-speaking people, the Subartians.
Then the Akkadians came along, conquered part of the Subartians, which included the city of Assur (Assyria). After that, the Assyrians became an Akkadian-speaking people. The other Subartian tribes migrated north, past Mount Ararat, to escape the Akkadians. That was the point in which the Assyrians and the Armenians became a separate people.
TL;DR: Assyrians and Armenians were one people that split shortly after the Akkadian invasions into Northern Mesopotamia.