in the public eye vs public

in the public eye is not a noun while public is a noun.

Word	Noun	Adjective	Verb	Adverb
in the public eye	No	Yes	No	No
public	Yes	Yes	No	No

in the public eye (noun)	public (noun)
	people in general considered as a whole
	a body of people sharing some common interest

in the public eye (adjective)	public (adjective)
of great interest to the public	not private; open to or concerning the people as a whole
	affecting the people or community as a whole