My understanding of these words is:
Infantile indicates behavior of very small children (infants), in contrast to immature that says someone is unexperienced and callow. Childish and puerile sounds to me as something between infantile and immature.

  1. In regular use, do these words allude any differences or are they used fully synonymic.

  2. Which word is most common?
    I've already taken a look on ngram, frequency seems to be quite equal.

  3. Which words are more formal or informal, which one are more or less colloquial?

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    what about 'juvenile'? – Mitch Jan 20 '12 at 22:26
  • Or 'jejune'...I started looking at Visual Thesaurus. – Mitch Jan 20 '12 at 22:47
  • Downvotes... For what reason? – Em1 Jan 20 '12 at 23:22
  • I once had a colleague whose motto was "You're only young once, but you can be immature forever." – MickeyfAgain_BeforeExitOfSO Jan 21 '12 at 2:17
  • 1
    @Em1: the downvotes, I'm guessing, probably had the same motivation as the close votes, that it is too basic a question. I answered because you're asking about the nuances among them that are not or cannot be extracted from dictionary entries. Even the (arguably) best dictionary (the OED) with its copious minutely dissected multiple definitions with examples and references cannot capture the full range of the nuances. – Mitch Jan 21 '12 at 18:09
  • There are no such things as exact synonyms, if that's what you intended by 'fully synonymous'. Natural language isn't mathematics; there aren't exact equals to replace. There's always some tiniest bit of nuance that will separate two different forms. Even words with the same form can have different meanings (see the dictionary definitions of the word 'set')
  • puerile is the least common in speech, infantile and juvenile next, then immature, then childish the most common (I have no ngrams for speech though so that's my personal recollection).
  • puerile is also the most formal, juvenile and infantile less so but still formal considering their association as technical terms, imature, then childish. But even though 'childish' will work in a colloquial setting it is not strictly colloquial and word work with no problem in a formal context (that is, it is not slang or vernacular at all).
  • as to their semantic differences, I'd call someone who tells fart jokes immature but not childish. And someone who (when too old for it) plays with dolls childish but not immature. Infantile I would use for an adult who gets upset when they don't get what they want. Puerile, juvenile, and immature seem pretty close except for their associations. I'm sure there are quite a few more distinctions that can be made and much further analysis beyond the examples I gave.
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You can follow the chain of synonyms for these words on Merriam-Webster Unabridged:

infantile : suitable to an infant : characteristic of an infant : very immature : CHILDISH

childish : marked by immaturity or simplemindedness : PUERILE, PETTY

puerile : unworthy of an adult : IMMATURE, CHILDISH

immature : exhibiting less than a normal or expected degree of maturity

None of these words are more or less formal, or more or less colloquial, than any of the others. Context is king, and the dictionary is your friend.

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When used as a pejorative each is more or less equivalent as insults need not be so specific. Used in other circumstances, one or two many me interchangeable depending on context, but they are not generally synonymous.

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