These terms mean "to make known to others what was intended to be kept secret, private, or confidential." Divulge is more likely to be used when something previously secret is revealed to a small number of people or a particular group; disclose usually refers to a general sharing of knowledge or information with others: "Barbara divulged to her roommates her plan to be married within a month." "The Warren Commission attempted to disclose the facts concerning President Kennedy's assassination." Neither word is widely used, perhaps because reveal, tell, and expose convey much the same meaning.
Dictionary of problem words and expressions. Harry Shaw. 1975.