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.
In regular use, do these words allude any differences or are they used fully synonymic.
Which word is most common?
I've already taken a look on ngram, frequency seems to be quite equal.
Which words are more formal or informal, which one are more or less colloquial?