cosmopolitan vs world-wide

cosmopolitan is a noun but world-wide is not a noun.

Word	Noun	Adjective	Verb	Adverb
cosmopolitan	Yes	Yes	No	No
world-wide	No	Yes	No	No

As adjectives, cosmopolitan and world-wide are synonyms defined as:

Other synonyms of cosmopolitan include ecumenical, general, oecumenical, universal, worldwide.

cosmopolitan (noun)	world-wide (noun)
a sophisticated person who has travelled in many countries

cosmopolitan (adjective)	world-wide (adjective)
growing or occurring in many parts of the world	of worldwide scope or applicability
composed of people from or at home in many parts of the world; especially not provincial in attitudes or interests	spanning or extending throughout the entire world
of worldwide scope or applicability	involving the entire earth; not limited or provincial in scope