Some words have the same etymology, root, but mean different things, such as mysterious and mystical. What are some other pairs (or more) that fall into this category, and what exactly is this category called?
There are actually lots of these. They are called doublets.
My favorite example, which is not mentioned on that Wikipedia page, is the following:
Genre, gender, generic, and genus all come from the Latin genus meaning "kind/class" (although the word actually goes all the way back to Proto-Indo-European *gen-/*gon-/*gn- meaning "produce, beget, be born").
Cognates with different meanings from different languages/dialects are known as False Friends.