The primary distinction would be that a kingdom, by definition, has a monarch. The United Kingdom has the Queen. The Kingdom of Saudi Arabia has the king, and so on and so forth.

Empires, on the other hand, do not necessarily have an emperor, rather, their defining feature is a single polity controlling other polities. So for example, the Roman Republic, before the time of Augustus, had already been an empire, just without an emperor, for centuries. The same thing applies to, say, the French Empire during the Third Republic, or American Empire. This also tends to apply to monarchies, so you have the German Empire, which was conceived of as an empire over and above its constituent kingdoms, or the British Empire, in which the United Kingdom ruled over various territories.

The one thing I'm not qualified to speak to is translations. So why is tenno in Japanese translated as Emperor while other terms are translated as king.

Sorry, meant to say "any" not "are."