If I understand Richard correctly, I think I agree with him. I see "childlike" as innocent behavior, like a child's. I see "childish" as an adult behaving like a child, which is inappropriate. Here's what the OED says on "childlike": "Like a child; (of qualities, actions, etc.) like those of a child, characteristic of a child. Almost always in a good sense, with reference to the innocence, meekness, etc., of children; opposed to childish, which is generally used in a bad sense."
Yeah, but what about this, which is pretty famous:
I remember, I remember The fir trees dark and high; I used to think their slender tops Were close against the sky: It was a childish ignorance; But now 'tis little joy To know I'm farther off from heaven Than when I was a boy.
My sense of childlike is not entirely positive. I realize it is supposed to mean innocent and unspoiled, but sometimes you see it used to describe an adult who is not quite all there. "She had an almost childlike quality" may mean that she was clueless or dense or a little slow--besides being unspoiled and kind or innocent in her nature.
As for childish, for me, that is always negative, even when applied to a child. Parents say to their children "don't be so childish" all the time, and the kids are by virtue of their age, childish. But when I hear the word, I think of selfish, tantrum-throwing, inconsiderate and rude as accurate synonyms.
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Another distinction I would make for "childish", especially when applied to an adult, is that it has to be a deliberately negative action – like a "childish" tantrum" or "childishly" refusing a request. Without the deliberate aspect, I would lean toward "childlike" as a description
As well as some of the descriptions on here, I also see 'childlike' as retaining positive aspects of being a child (playfulness, etc), and 'childish' as retaining negative ones (e.g. sulking, blaming others, refusing to accept responsibility for one's actions etc).
For example, I'm 33 and still love making snowmen (on the rare occasions we get snow!), and sand sculptures on the beach, and a lot of my friends do too. I feel sorry for people who believe that when you hit a certain age you're not supposed to enjoy certain things anymore, and that if you do enjoy them, you're certainly not supposed to engage in them. If it causes no harm, why deprive yourself? I also enjoy and engage in more 'mature' pursuits, but if I want to paddle in the sea and make a Sand Dalek, I will.
I think one of the reasons people have kids is so that they can still enjoy those sorts of pursuits, as it's more acceptable to them because they're just playing with their children .
Do you know, I've never made a snow angel. I've been planning to the last few years but it never snows enough anymore. This year we've had two snowfalls of a couple of inches and that's it. Most annoying.