modern world vs contemporary world

modern world and contemporary world both are nouns.

Word	Noun	Adjective	Verb	Adverb
modern world	Yes	No	No	No
contemporary world	Yes	No	No	No

As nouns, modern world and contemporary world are synonyms defined as:

modern world and contemporary world: the circumstances and ideas of the present age

Other synonyms of modern world include modern times, present times.

modern world (noun)	contemporary world (noun)
the circumstances and ideas of the present age	the circumstances and ideas of the present age