Childish is generally a bad thing. It usually refers to an adult who exhibits the bad qualities of being a child. For example, crying because you did not get candy is childish. Childlike is generally a good thing. It refers to the good aspects of children. So, you would not say "I know he is popular but I find him childlike". You would say "I know he is popular but I find him childish". Note that you usually do not need to explain how someone is "childish" but you do need to explain how they are "childlike" because it describes a characteristic of a person, not the person himself. For example, I could say "He is childish", and most people would understand what I mean. If I told them "He is childlike", they would be likely to ask "What do you mean specifically?" or "In what way?" Instead, you should say something like "The way he tries to be friends with everyone is childlike". In this case you are not directly describing him, you are describing "the way he tries to be friends with everyone".

childish: Он ведет себя по-детски, несерьёзный, ребя́ческий childlike: Он выглядит как ребенок, де́тский - как у ребёнка