Both adjectives refer to the characteristics of a child. Childish--which means silly or immature--generally points to unfavorable qualities. Childlike--which means trusting or innocent--refers to more favorable qualities.

Link to present incidents:
'For goodness' sake, grow up and stop being so childish!'
 'He had an almost childlike trust in the goodwill of his friends.'
From: https://www.thoughtco.com/childish-and-childlike-1689341

Please, ignore the headline.
No reference to current context for a while - sorry!