is there a difference between king and emperor?

To understand the difference, one must consider the historical contexts when the terms come into being.

Generally, the term kingdom is used when the inhabitants were largely homogenous in ethnicity, language or culture, and also share much common heritage.  Certain kingdoms contain pockets of regional minorities, and there is a tendency for these minorities to be assimilated by the dominant ethnic group.

Examples: Historical kingdoms of England, Scotland, Wales, Ireland, France, Italy, Greece, Denmark, Sweden etc.

The term empire is used when the inhabitants were heterogenous in ethnicity, language or culture, and had been merged from previously separate states with distintively different people.  Though the empire is formed by a dominant ethnic group, the sheer vastness of the empire and diversity meant that the heterogenous demography would remain.

Examples: Historical Roman Empire, Byzantine Empire, Holy Roman Empire, Russian Empire.  Napoleon could only be called an emperor properly when he conquered non-French territories.

The term 'emperor' is also applicable to a king who conquered other sizable kingdoms.

Historical background (Europe)

In Europe, the term 'emperor' originated with the ancient Romans.

The Romans found themselves overlords of a diverse group of people, from Celts and Greeks in Europe to Egyptians in Africa to Arabs and Jews and Pheonicians in the Middle East.  There was no way everyone would take up Latin and drop their own culture.

Historical background (China)

In China, prior to the Spring Autumn and Warring States period, regional languages existed.  The Empire was formed when the State of Qin conquered all other states, ruling a territory many times larger than any kingdoms China had ever known.

Through strong institutional measures, assimilation was carried out successfully in China.

In China, after uniting all the other kingdoms, Qin Wang (the King of Qin) entitled himself "Huang Di (Emperor)" - a newly coined term which means his achievements surpassed those of "Huang"s or "Di"s not to mention "Wang (King)"s.

There was no king all through Qin Empire, but before surrender to Liu Bang, the third emperor of Qin changed his title back to king, showing he was no longer the ruler of all China.

From Han on, emperors gave the titles of king to their brothers and some generals/officials as highest honors. Rulers of Chinese vassals were often given the titles of king too.

So emperors were considered superior to kings. Korea, being a Chinese vassal (most time), its rulers had the titles of king. Were they called emperors, it would have caused political troubles with China.

As for Japan, though greatly influenced by China, it had never been a Chinese vassal. Hideyoshi Toyotomi was once titled "the King of Japan" by the Ming Emperor. But Japanese rulers sought every chance to show they were equivallent to Chinese emperors.

Back to the recent centuries, the Korean King was irritated by the Japanese Tenno (as "Emperor") calling him "King" in his letters (I saw it from TV series).