It has to do with the periodization of ancient Near Eastern history. Assyrian history is divided into the Old Assyrian, Middle Assyrian, and Neo-Assyrian periods, corresponding approximately to the Middle Bronze Age, Late Bronze Age, and Iron Age, respectively. Since Assyria acquired its empire in the Neo-Assyrian period, it is called the Neo-Assyrian empire.
The major city of the Assyrians was Aššur in Upper Mesopotamia. Excavations reveal that the site was occupied in the Akkadian period (ca. 2350-2150 BCE) and would have been subject first to the Akkadian kings and then the Ur III kings. (For more information, see the Met Museum's website for a free PDF of Assyrian Origins: Discoveries at Ashur on the Tigris). After the collapse of the Ur III state, Aššur flourished in the Old Assyrian period (ca. 1975-1775 BCE). The Old Assyrian period is best known for its trade network in which merchants from Aššur traveled to Anatolia via donkey caravans to exchange textiles for gold and silver. Many Assyrians eventually settled down in Anatolia, married Anatolian women, and produced children.
Several factors contributed to the collapse of the Old Assyrian trade network, including warfare in Syria and Upper Mesopotamia and the coalescence of the Hittite kingdom in central Anatolia. The Assyrian king Šamši-Adad I created a rather sizable kingdom in Syria and Upper Mesopotamia centered at Šubat-Enlil (Tell Leilan), but most of his territorial gains were lost by his son Išme-Dagan, who was overshadowed by the aggressive expansionism of Hammurabi of Babylon. The growth of the powerful state of Mitanni by 1500 BCE prevented Assyria from regaining prominence for a couple of centuries.
It was only with the weakening of Mitanni that Assyria once again rose to prominence. The Assyrian king Aššuruballit I wrote to the king of Egypt as a "brother," which irritated and concerned the Babylonian king Burnaburiaš II, who considered Assyria one of his vassals. In one of the Amarna letters (EA 9), the later wrote:
Now, as for the Assyrian, my vassal, it was not I who sent him to you. Why on their own initiative have they come to your country? If you love me, they will conduct no business whatsoever. Send them off to me empty handed!
Babylonia lacked the power to contain Assyria, however, and it quickly became one of the Great Powers of the Late Bronze Age. After he defeated Mitanni, the Assyrian king Adad-nirāri I wrote the Hittite king as a brother, and the latter wrote a strongly worded rebuff (KUB 23.102).
You keep speaking about the defeat of Wašašatta and the conquest of the land of Ḫurri. You conquered by force of arms. You conquered by .... So you’ve become a “great King,” have you? But why do you continue to speak about “brotherhood” and about coming to Mount Ammana? What is it, (this) “brotherhood”? And what is it, (this) “coming to Mount Ammana”? For what reason should I call you my “brother”? Who calls another his “brother”? Do people who are not on familiar terms with each other call each other “brother”? Why then should I call you “brother”? Were you and I born of the same mother? As my grandfather and my father did not call the King of Assyria “brother,” you should not keep writing to me (about) “coming” and “Great Kingship.” It displeases me!
The Middle Assyrian state had its last and most notable victory during the reign of Tukulti-Ninurta I, who conquered the city of Babylon. Assyria, like Egypt, was weakened by the collapses of the Late Bronze Age but survived. Assyria briefly expanded once again during the reign of Tiglath-Pileser I (1114–1076 BCE), who regained much of Syria and Upper Mesopotamia. Little can be said of the next 120 years or so of Assyrian history, however. The rise of the Aramaeans in the 1st millennium BCE and the coalescence of the Neo-Hittite and Aramaean city-states disrupted Assyria's territorial expansion, and by 950 BCE, Assyrian power was at an ebb.
This state of affairs changed with Aššur-dan II (934–912 BCE), variously considered the last king of the Middle Assyrian period or the first Neo-Assyrian king. The next century (934–824 BCE) saw Assyria regain most of its territory. By the reign of Aššurbanipal (668–631 BCE), the Neo-Assyrian empire was at its height, having conquered Egypt, the Levant, Arabia, Mesopotamia, and parts of southern Anatolia.
This tripartite division of history is pretty common in Near Eastern archaeology. Egyptian history is divided into the Old, Middle, and New Kingdoms (with intermediate periods between each). Arthur Evans, following this Egyptological convention, divided the history of Crete into the Early, Middle, and Late Minoan periods, followed by the adoption of the terms Early, Middle, and Late Helladic (for mainland Greece) and Early, Middle, and Late Cycladic (for the Cyclades). Modern scholars have revised the divisions for Crete somewhat, and Prepalatial, Protopalatial, Neopalatial, and Postpalatial are now preferred. Babylonian and Elamite history are likewise divided into the Old Babylonian, Middle Babylonian (Kassite), and Neo-Babylonian periods and Old Elamite, Middle Elamite, and Neo-Elamite periods, respectively. As for the Hittites, Hittite linguists and historians are still at odds over whether one can speak of a "Middle Kingdom" in Hatti.