What are the differences between the Han and Roman Empires?

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.