Childlike means to behave like a child. It refers to the temperament of the person. Those good qualities which are seen in children such as honesty, faith etc are often referred to as childlike. Childlike is used as an adjective.
Childish means silly, immature, foolish. It refers to the behavior of a person. Childish behavior are considered bad characteristics of an adult. It is not expected from an adult. This is also used as an adjective.
Usage of Childlike:
1.Sita has a childlike simplicity
2.He has a childlike innocence in his relations.
3.She believes others in a childlike way.
Usage of Childish
1.Ramu's conduct is extremely childish.
2.He was very childish to speak rude to the guest.
3.He ruined the party in a childish manner.