I believe that the reason we translate tennou (天皇) as emperor is because it contains the character for emperor; which is incidentally the same character as the Chinese use to denote emperor, 皇. Basically the Japanese were using the more exalted character for their supreme ruler. If they had used the less exalted character, 王 (king) we would probably refer to the Japanese ruler as a king.

Basically English follows the Japanese (and Chinese) lead. The used the character for emperor, so we followed suit and also called the Japanese rulers, emperors.

In early contact reports by Europeans, they generally refer to the daimyo as kings. Therefore, a ruler over many kings should be an emperor. Once that became the translation, it stuck.

Japan is certainly not the only nation where an obsolete political seat continues to be carried on based on tradition. It's not much different from the royal family of Britain. As for why we continue to use the term "emperor", again, tradition. Why change the term everyone already knows?

It's worth noting, though, that Douglas MacArthur was a major proponent of keeping the seat of the emperor after the war, and specifically letting Hirohito remain emperor rather than being tried as a war criminal. He felt it was important to the national identity and stability of Japan to retain the throne. So, had things been handled differently, perhaps there wouldn't be an emperor today.

*Edited for clarity.