The two kingdoms are not identical. Although there is some overlap in geographical area, the kingdoms existed in different periods and had different languages. Sumer originated in southern Mesopotamia and Assyria in northern Mesopotamia. Both kingdoms marked important phases in the development of the Mesopotamian civilization, or the area "between the two rivers" of the Tigris and Euphrates in what is now Iraq. They did share many features in common, including basic social structure, use of cuneiform in writing, and some religious and artistic traditions.

Sumer was the smaller and earlier of the two kingdoms, responsible for, among other things, the invention of writing ca. 3500-3000 BC. The Sumerian tongue is a language isolate, with no surviving close relatives. However, due to the cultural prestige of Sumer and its early development of writing, Sumerian language, myth, and literature had an important influence on all subsequent Mesopotamian languages and cultures. City states arose in Sumer ca. 6500 BC and flourished until it was conquered by northern Mesopotamian powers ca. 2000 BC. Most students know Sumer through the Epic of Gilgamesh, which tells the story of a quasi-legendary king of the great Sumerian city-state Uruk.

The Assyrian Empire arose in northern Mesopotamia, and derives its name from the important city of Assur. The Assyrians spoke Semitic languages but adopted cuneiform writing from Sumer. It first flourished as the Akkadian (sometimes called neo-Sumerian) Empire and continued through ca. 900 BC, collapsing and arising again as the powerful and extensive neo-Assyrian empire which lasted from 911 to 609 BC.

No, the Sumerians and Assyrians were not the same civilization (though they both inhabited Mesopotamia). The Sumerians were far earlier, dating back as early as 4000 BCE, and can themselves be subdivided into various city-states (which were often in conflict with one another).

The Assyrians arose much later, originating perhaps around 2000 BCE. The Assyrians reached the apex of their power between the ninth and seventh centuries BCE, when they conquered Mesopotamia, as well as large sections of the Middle East.

The thing to keep in mind is that Mesopotamia actually saw a number of civilizations rise to power in the region. For example, one can also point towards Sargon and the Akkadians (who conquered the Sumerian City States in around 2300 BCE, though that empire collapsed within about a century of Sargon's death), as well as the Babylonians, another major civilization and (at times) empire-builder in the region. One can additionally look towards conquests from outside Mesopotamia altogether, as seen by the examples of Persia, or (later) Alexander the Great.