Are Japanese and Chinese people related at all?

Yes. However, the people of Japan are more closely related to the people of the Korean peninsula. Many early immigrants came from Korea to Japan.

If yes, why do they speak completely different languages?

"Related" doesn't mean "the same". Italian and Greek people are related; they don't speak the same language. Hell, Italy and Greece and geographically connected and they don't speak the same language.

If not, why do they share so many cultural similarities?

Much like how Rome was considered to be the height of civilization in the Western world, China was considered the height of civilization in East Asia. The written language of China in particular was considered to be a mark of civilization; much like Latin in Europe. Many things like metal working, writing, religious traditions, etc, came from China and was eventually given to the Japanese by the previously mentioned Korean immigrants.