The Han and Roman Empires were contemporaneous with one another but situated in opposite sides of the world. There was trade carried on between them (mainly along the Silk Road) but the Han Dynasty was situated in China while Rome was situated in the Mediterranean. This meant there was a vast array of differences between the two political states, in that they evolved largely in isolation from one another.

The Han had their origins in a rebellion against the earlier Qin Dynasty. The Qin was a short lived dynasty, created out of the unification of China under the first Emperor, ruled under legalist principles. Much of Han politics were shaped in reaction to the harsh and suppressive policies of the Qin, intending to replace legalist principles with Confucian ones. For example, the Han oversaw the implementation of the Civil Service Exams which would remain a cornerstone of Chinese governance for centuries afterwards.

The Roman Empire expanded across the Mediterranean and politically evolved over time from a Republican form of government which, through the instability of the late Republic, was ultimately replaced by the Imperial power structure dominated by the Emperors. Rome did not really have an equivalence to the Confucian Scholar-bureaucrats. Religiously the two were also very different from one another. One of the most lasting influences Rome had on European culture was the establishment and growth of Christianity, which was eventually made the State Religion of the Roman Empire.

These only represent a few of the differences between these two States. Ultimately, we must recognize that, when we speak of distances as vast as those between the Mediterranean and East Asia (especially in the pre-industrial world) we'd be looking at profound differences in cultural, social, and political contexts. The potential avenues of exploration go far beyond the limited and very basic sketch established here.

In what ways were the Roman empire and the Han empire similar and in what ways were they different?

The Han dynasty in China maintained a remarkable level of similarity with its predecessor dynasty, the Qin, founded by the first emperor Qin Shi Huangdi. The Han dynasty borrowed many traditions from the Qin, such as the creation of a highly centralized state, a systematic bureaucracy, a large military, and a unified system of writing. With these very effective tools of statecraft, Han emperors claimed to rule “Under All Heaven”—Tianxia—which to them was literally all of the known world. This idea fell under the general rubric of the Han ideology known as the “Mandate of Heven,” by which each successive Han emperor claimed that heaven had given them the right to rule all of Tianxia, the previous emperor having lost that right.

In many important ways, the Roman empire demonstrated similarity to this style of rulership. With the advent of Octavian to a position of unprecedented power in the Senate, the republican period ended, and the Roman empire was officially born in 27 BCE. Octavian, and all of the emperors that came after him, claimed to rule through imperium, which literally means “command” but was usually applied in military sense. This bore similarity to (but was not the exact same thing as) the concept of the Han dynasty's Mandate of Heaven. Some Roman philosophies also bore strong resemblance to Chinese ways of thinking during the same period. Hellenistic Stoic philosophers such as Epictetus, for example, claimed that the entire Mediterranean basin was one large “ecumene” because all of its inhabitants shared a common Greek language and way of thinking. In this way, the Stoic (and other) philosophies that allowed Romans to mentally unify their empire were similar to Han conceptions of Tianxia that served similar functions.

One critical way in which these empires were different was their conceptualization of power. The Roman empire had been built on the principles of res publica—the “public thing.” The idea of res publica implied that even when the emperors held the most dictatorial powers in the Senate, this institution was still a representation of the collective demands of the citizens of the empire. In practice, of course, most ordinary Roman citizens were powerless. But this very idea conflicted with the Han understanding of the Mandate of Heaven, which was by its very nature a private blessing. The Mandate that was bestowed on a Han emperor came from the sky, and it was a reward for virtuous and ethical behavior. Though anyone could theoretically gain heaven’s favor, once an individual did obtain the Mandate and began his rule, it was his alone.

In what ways were the Roman empire and the Han empire similar and in what ways were they different?

Both were major, long-lasting regional empires. While the Han dynasty lasted for around 2,000 years (starting around 206 BCE), the Roman Empire was slightly older and shorter-lived, lasting from 753 BCE to around 476 CE. In the first century CE, both had similar populations: usually estimated in the 50–60 million range. Both of these empires are often looked back upon as representing a "golden age" of early civilization as well. It is also worth noting that both were deeply hierarchical societies.

Perhaps because of its location and earlier fall, Roman law laid the foundation for much of the legal structures of the Western world after the renewed interest in Greco-Roman thought during the Renaissance.

Ultimately, though, this example starts to show the limits of this kind of comparison. The Roman and Han empires are, quite simply, different empires occupying different temporal and spacial locations; thus, their influence is wildly different. The Roman Empire is a huge point of reference for European and European-based society, and the Han dynasty occupies a very different role.