ballpoint pen vs ballpen

ballpoint pen and ballpen both are nouns.

Word	Noun	Adjective	Verb	Adverb
ballpoint pen	Yes	No	No	No
ballpen	Yes	No	No	No

As nouns, ballpoint pen and ballpen are synonyms defined as:

ballpoint pen and ballpen: a pen that has a small metal ball as the point of transfer of ink to paper

Other synonyms of ballpoint pen include ballpoint, Biro.

ballpoint pen (noun)	ballpen (noun)
a pen that has a small metal ball as the point of transfer of ink to paper	a pen that has a small metal ball as the point of transfer of ink to paper