pseudonym vs pen name

pseudonym and pen name both are nouns.

Word	Noun	Adjective	Verb	Adverb
pseudonym	Yes	No	No	No
pen name	Yes	No	No	No

As nouns, pen name is a hyponym of pseudonym; that is, pen name is a word with a more specific, narrower meaning than pseudonym:

pseudonym: a fictitious name used when the person performs a particular social role
pen name: an author's pseudonym

Other hyponyms of pseudonym include stage name, nom de plume, hagionym.

pseudonym (noun)	pen name (noun)
a fictitious name used when the person performs a particular social role	an author's pseudonym