What is the difference between someone childlike and someone childish?

Yes! it's like the difference between 'uninterested' and 'disinterested'.

Childlike means a person has a playful outlook on life, that he likes what a child would like. We are all a little childlike.

Childish means acting like a child. Demanding things, lying as children sometimes do, blaming others for their mistakes and failures, not taking things seriously like an adult. Like a certain president of the US that I could mention.

I associate childlike with innocence and inexperience. I associate childish with defiance, inappropriate behavior and selfishness. I expect someone who is behaving childishly to know better.

Childish behaviour is fine in a child but in an adult its inappropriate as it means the adult has not grown up and is immature, overreacting, headstrong, defiant, impulsive, etc, etc. If you see an adult throw himself on the floor and spit the dummy because the shop doesn't have the lollypop he wants, it would look quite stupid, but if you think about it many adults do in fact behave like this in various situations, speeding fine, road rage, not getting their own way, using intimidation and anger etc, This is all characteristic of someone who has not learned to govern their emotions, its a sign on weakness of character.

Child like meaning having the characteristics of a child an emotional innocent normal behaving child. Childish describes the petulance and immature behavior, guided essentially by selfish, unresponsive behavior lacking knowledge, and appropriateness acting out temper tantrums with no genuine concerns for whom you interact with.

Childish is almost always used as an insult- usually referring to someone stubborn, who always wants their own way and will have a "tantrum" if they don't get it, stupid often selfish.
Childlike usually refers to someone curious, innocent, naive. Childlike is usually used in a positive way- referring to the good qualities of a child.