private corporation vs privately held corporation

private corporation and privately held corporation both are nouns.

Word	Noun	Adjective	Verb	Adverb
private corporation	Yes	No	No	No
privately held corporation	Yes	No	No	No

As nouns, private corporation and privately held corporation are synonyms defined as:

private corporation and privately held corporation: a corporation owned by a few people; shares have no public market

Other synonyms of private corporation include close corporation, closed corporation.

private corporation (noun)	privately held corporation (noun)
a corporation owned by a few people; shares have no public market	a corporation owned by a few people; shares have no public market