The two words both translate to "winter" in English.1,2
Although information on Wikipedia is sparse, I gather that hiems is most commonly used as a noun, while hibernum is the noun form of the more common adjective hībernī (wintry).
Both forms, however, are shown as nouns. How do they differ in usage? Are they essentially interchangeable synonyms?
As you noted (almost correctly), hibernum comes from hibernus, -a, -um which does mean 'wintry'. Any neuter adjective can be made into a noun by simple use. It's a feature of the language.
However, hibernum is rare and not often used. 99% of the time when an author wants to say 'winter', they say hiems, if not perhaps some poetical euphemism. Hibernum's importance comes not from meaning winter, but rather from its meaning in the plural as 'winter-quarters for troops". See Lewis and Short II.B.
