The full answer to this question would truly be a rather vast body of work and I advise you to do some research of your own in this regard. But I will give you a summary of the most prominent points.
The Han dynasty of China was the 2nd imperial dynasty. It existed from 206BCE to 220CE, approximately 425 years and was considered a golden age in China's history.
The Roman "empire" existed, strictly speaking, from 31BCE until 476CE. The whole duration of the Roman world existed far longer i.e. 509-31BCE [Republican era]; 31BCE-476CE [Imperial era]; 395-1453CE [Byzantine or Eastern Roman empire].
Han China: Well organized bureaucracy based upon Confucian ideas and education.
Roman Empire: Well organized bureaucracy founded on Roman law and classical learning.
Han China: Emphasis on family ancestors and patriarchy; reliance on landed gentry.
Roman Empire: Emphasis on family and the pater familias; reliance on patricians.
Han China: Engineering accomplishments: roads, canals, the Great Wall.
Roman Empire: Engineering accomplishments: roads, aqueducts, amphitheatres, domes, sewage systems, central heating and the Colosseum.
Han China: Religion: Confucianism, Daoism, native gods, introduction to and start of Buddhism.
Roman Empire: Religion: diverse pagan pantheons, emperor as a god, introduction to and start of Christianity.
Decline of Han Dynasty: Infighting among ruling elites; inequitable distribution of land [i.e. tax burden fell on peasants rather than on large landowners]; series of peasant rebellions; generals usurp political power and became warlords; 220CE generals divide empire into three kingdoms; emigration of nomadic peoples into North China kept the country divided.
Decline of Roman Empire: internal opposition [barrack emperors]; generalissimos in the 3rd century CE; agrarian crisis and slave labour; difficulties in administering such a vast empire creates rivalries and divisions of authority; division of empire into Eastern and Western empires; capital moved to Constantinople; Germanic invasions by Vandals, Ostrogoths, and Visigoths; 476 Odaecer deposes final Western Roman Emperor; Eastern Roman Empire becomes Byzantine Empire.
Shared Characteristics of Decline of Han China and the Roman Empire: decline in morals and values; decline in those values that have held the particular society together; public health and urban decay; political corruption; unemployment and inflation due to excessive use of slave labour; inadequate technological advances; military spending and corruption.
Both empires were agricultural based with homogenous cities and diverse peoples.
Religion was perhaps the prime distinction between the two nations. While the Han appeared to benefit from the doctrines of Confucius, Christianity was accepted by the Romans, but only after its acceptance from Constantine. This religion eventually influenced many decisions and beliefs of future rulers of Rome. Buddhism, a heresy to Confucius idea, was just emerging in China and did not entice the majority of the population until the post-Han era.
Two books on the subject that you can consult are:
Scheidel, Walter. Rome and China: Compariative Perspective on Ancient World Empire [Oxford Studies in Early Empires]. published by OUP, USA. 2009.
Miller, Frederic P; Vandome, Agnes F; McBrewster John. Comparison between Roman and Han Empires. published by Alphascript Publishing. 2009.