Various synonym guides through the years have attempted to delineate the overtones of these three terms. George Crabbe, English Synonymes Explained, in Alphabetical Order (1816) discusses childish in company with infantine and then turns his attention to juvenile and puerile in relation to the heading term "Youthful":
Childish, Infantine. Childish is in the manner of a child. Infantine is in the manner of an infant. What children do is frequently simple or foolish; what infants do is commonly pretty and engaging, therefore childish is taken in the bad, and infantine in the good sense. Childish manners are very offensive in those who have ceased according to their years to be children. The infantine actions of some children evince a simplicity of character.
...
Youthful, Juvenile, Puerile. Youthful signifies full of youth, or in the complete state of youth; juvenile, from the Latin juvenis, signifies the same; but puerile from puer a boy, signifies literally boyish. Hence the first two terms are taken in an indifferent sense; but the latter in a bad sense, or at least always in a sense of what is suitable to a boy only; thus we speak of youthful vigor, youthful employments, juvenile performances, juvenile years, and the like; but puerile objections, puerile conduct and the like. Sometimes juvenile is taken in the bad sense when speaking of youth in contrast with men, as juvenile tricks; but puerile is a much stronger term of reproach, and marks the absence of manhood in those who ought to be men. We expect nothing from a youth but what is juvenile; we are surprized and dissatiusfied to see what is puerile in a man.
Almost a century later, James Fernald, English Synonyms and Antonyms, thirty-first edition (1914) offers this discussion under the heading "Youthful":
Boyish, childish, and girlish are used in a good sense of those to whom they properly belong, but in a bad sense to those from whom more maturity is to be expected; childish eagerness or glee is pleasing in a child, but unbecoming in a man; puerile in modern use is distinctly contemptuous. Juvenile and youthful are commonly used in a favorable and kindly sense in their application to those still young; youthful in the sense of having the characteristics of youth, hence fresh, vigorous, light-hearted, buoyant, may have a favorable import as applied to any age, as when we say the old man still retains is youthful ardor, vigor, or hopefulness; juvenile in such use would belittle the statement.
Like Crabbe, Webster's Dictionary of Synonyms (1942) addresses the three terms in two separate entries—childish under the heading "childlike" and juvenile and puerile under the heading "youthful":
Childlike, childish agree in meaning of having or showing the manner, spirit, or disposition of a child Both are applicable to adolescents and to adults as well as to children. Childlike, however, usually suggests those qualities of childhood which are worthy of admiration or emulation, such as innocence, simplicity or straightforwardness; childish suggests its less pleasing and less admirable characteristics, such as helplessness, peevishness, or undeveloped mentality; ...
...
Youthful, juvenile, puerile, boyish, virgin, virginal, maiden come into comparison only when they mean of, pertaining to, or characteristic of, one who is between childhood and adulthood. They are, however, not synonyms; for, although their basic meaning is the same, they are seldom interchangeable, because of widely differing implications and applications. ... Juvenile often suggests immaturity of mind or body or lack of experience; it is therefore applied especially (but not exclusively)to that which pertains to, or is suited to or designed for, boys and girls in their teens; as juvenile dances; juvenile fiction; a juvenile performance. Puerile is almost exclusively applied to acts, utterances, and the like, which are excusable in a boy or girl or are characteristic of immaturity, but are unpardonable or out of character in an adult: the word is now used chiefly in contemptuous reference to acts or utterances of the mature.
S.I. Hayakawa, Modern Guide to Synonyms (1968) categorizes all three terms under the heading "childish":
childish, childlike, immature, infantile, juvenile, puerile These words refer to behavior unbecoming to an adult. Childish suggests lack of control; or restraint and carries possible connotations of petulance, irrationality, impatience, and self-preoccupation: a childish insistence that everyone listen to him; a childish terror of insects. The word is so negative in tone that it is seldom used now used to describe a young person, or even his understandable limitations. ...
Unlike the foregoing immature and juvenile may be used in factual reference to the young or things pertaining to them: an immature seedling; a juvenile court. ,,, In reference to adults, both words are disapproving in tone, juvenile somewhat less and immature considerably less than childish. {I suppose we are all a little immature, but while I have been guilty of juvenile behavior from time to time, he is downright childish.} Juvenile indicates behavior typical of a young person nearing adulthood, and thus may suggest intemperance, extremism, laziness, excessive zeal or idealism, or thoughtless inconsiderateness: a juvenile inability to compromise; a juvenile habit of seeing everything in black and white; juvenile snap judgments. ...
Infantile and puerile are the most disapproving of all these words. From their construction, both might be thought capable of neutral description, but puerile is almost never used in this way, and infantile only when used as a technical word; ... The most formal of these words, puerile is also more vague in its condemnation. It can suggest anything callow, weak, stupid, or inept. At its most vague, it can mean simply worthless: puerile efforts to negotiate; the pianist's puerile performance.
Discussion
Hayakawa's analysis of infantile, childish, and juvenile makes intuitive sense as a mapping of hierarchical maturity (or the lack thereof) on people who are old enough to know better. In his reading, calling someone's behavior infantile is worse than calling it childish, which is worse than calling it juvenile—simply because infantile properly pertains to babies, childish to young children, and juvenile to teenagers.
It is interesting, however, to see Crabbe, back in 1816, dispute any such hierarchical or progressive understanding of the terms, evidently on the grounds that babies are cuter than children. Something similar may be at work in the ordering that Ann Gaffney applies to juvenile and childish in her answer to this question, with children beating out teenagers in the cuteness derby. There is, separately, some possibility that the connotations of juvenile as applied critically to the behavior of people who should be acting their age have become more pejorative as a result of the extended association of juvenile with the term "juvenile delinquent," although I'm not aware of any practical way to test that hypothesis.
As for puerile, it has been under a cloud for at least two centuries, despite the fact that its Latin root puer (child) is not inherently deplorable. Even in 1816, Crabbe notes, "puerile is a much stronger term of reproach" than juvenile. And all subsequent comparisons affirm the deeply disapproving sense in which speakers and writers typically use puerile.