public eye vs limelight

public eye and limelight both are nouns.

Word	Noun	Adjective	Verb	Adverb
public eye	Yes	No	No	No
limelight	Yes	No	No	No

As nouns, public eye and limelight are synonyms defined as:

Other synonyms of public eye include glare, spotlight.

public eye (noun)	limelight (noun)
a focus of public attention	a lamp consisting of a flame directed at a cylinder of lime with a lens to concentrate the light; formerly used for stage lighting
	a focus of public attention