cologne vs eau de cologne

cologne and eau de cologne both are nouns.

Word	Noun	Adjective	Verb	Adverb
cologne	Yes	No	No	No
eau de cologne	Yes	No	No	No

As nouns, cologne and eau de cologne are synonyms defined as:

cologne and eau de cologne: a perfumed liquid made of essential oils and alcohol

Other synonyms of cologne include cologne water.

cologne (noun)	eau de cologne (noun)
a perfumed liquid made of essential oils and alcohol	a perfumed liquid made of essential oils and alcohol