Historically, mizrachi Jews' ancestors followed different migration patterns in antiquity. Under the Roman Empire, many Jews migrated throughout the empire, including to North Africa and Syria. There was also an existing community in modern-day Iraq, from the Babylonian Exile era. This community was among the largest and most prestigious for much of Jewish history.

As for genetics, most studies indicate that both share similar Levantine Y-chromosomes, but Ashkenazim have more European maternal lineage. Studies differ on this though, and and you'd need a geneticist to sort out how these studies vary.

As for appearance, keep in mind that a tiny rate of mixing adds up over generations. One non-Jew converting in a tons each generation, for instance, ends up with a high rate of non-Jewish ancestry after a long enough time. Also, there are plenty of relatively dark-skinned Ashkenazim, and relatively light-skinned mizrachim. /r/askscience would be a good place to ask about whether or not ~2000 years is long enough for skin color to change without any population-mixing.