Childish - relates to something typical of a child, often in negative sense: childish games, childish fears, high, childish voice. It has negative connotations. It´s often synonymous with words infantile, immature, silly, foolish.
'For goodness' sake, grow up and stop being so childish!'

Childlike - qualities associated with the ideal child, such as innocence, charm, beauty, trust: childlike charm, childlike trust. it doesn´t usually have negative connotations. It´s closer to words innocent, trusting, naive, pure which aren´t negative.

Fowler on Childish and Childlike
"The distinction drawn is so familiar that childish is in some danger of being restricted to the depreciatory use that is only one of its functions, while childlike is applied outside its sphere; the face, for instance, that we like a child to have should be called not a childlike, but a childish face; the rule that childish has a bad sense is too sweeping and misleads. Childish used of adults or their qualities, and childlike (which should always be so used), have the opposite implications of blame and approval; childish means 'that ought to have outgrown something or to have been outgrown,' and 'childlike 'that has fortunately not outgrown something or been outgrown'; childish simplicity in an adult is a fault; childlike simplicity is a merit; but childish simplicity may mean also simplicity in (and not as of) a child, and convey no blame; childish enthusiasm may be either a child's enthusiasm or a man's silly enthusiasm; childlike enthusiasm is only that of a man who has not let his heart grow hard."
(H. W. Fowler, A Dictionary of Modern English Usage: The Classic First Edition, 1926. Ed. by David Crystal. Oxford University Press, 2009)

childISH vs childLIKE