I'll have to go with the Romans on this one, for the same reason I chose the Vikings in the Vikings vs. Spartans thread: the deciding factor being the superior techniques and technology of a later time period.
The Assyrians certainly had the largest empire the world had yet seen up until their time. However, the Roman Empire came nearly a thousand years later, and by then the Romans had techniques and technologies that would have been superior to the ones used by the Assyrians in their time.
This is why I don't think it's a fair comparison to be pitting two armies from completely different time periods against each other, because it usually ends up giving the ones from a later time period an unfair advantage over those from an earlier time period.
Assyrian would of destroy roman easily ,assyrians had powerful military , plus they were all smart , know how to handle horses very well , they had powerful weapons , also they were the first ones to make iron weapons , also there kings were really smart , assyrians were way before romans so it wouldn't happen They fought with chariots, on foot, horseback, and had good iron weapons. They were hated and feared.