Research some of the more instrumental pieces of art from both ancient cultures. The University of Chicago's Oriental Institute offers collections of ancient Mesopotamian art whereas the University of Memphis offers information and photographs of ancient Egyptian pieces.
Analyze trends in ancient Mesopotamian art pieces. Many of the pieces depicted on the University of Chicago website consist of figurines. The first piece listed on the site is a figurine of King Ur-Nammu, who enriched one of the temple, or E-kur, of Enlil, one of the most important gods. There are also figurines of a random female and statuettes of Sumerians kneeling in front of gods.
Compare and contrast this with similar ancient Egyptian art pieces. Ancient Egyptian art often depicted Egyptians and Egyptian kings as well, but on a larger scale. Ancient Egyptian kings, such as Ramses II, are often depicted in larger-than-life statues outside of temples, such as at the temple at Amarna. The statuette of Sumerians kneeling share one similarity with the statues of Egyptian art: the eyes are large and rimmed in black.
Compare and contrast the temples built by ancient Mesopotamians and ancient Egyptians. Fairfield University shows pictures of Sumerian temples on its site dedicated to ancient Mesopotamian architecture. Sumerian temples contained an outside retaining wall and stairs leading up to a wide plaza and a temple building, much like the ancient temples in Jerusalem. Ancient Egyptian temples, by contrast, tended to sport large statues of pharaohs at the entrance and then rows of columns inside.
Note that both ancient Mesopotamian art and ancient Egyptian art contained steles with depictions of people or animals. The University of Chicago's website offers a photo of a stele of a lion that adorned the Processional Way out of Babylon. The ancient Egyptians also painted reliefs on columns and in the tombs of their pharaohs.
Observe that pyramids are unique to ancient Egyptians; though there are pyramids in ancient Mesoamerican cities, the ancient Mesopotamians did not build pyramids.