If I told you a mutual acquaintance of ours had criticised your new hairdo, you might well take offence. But if I said I’d be happy to criticise something you’d written, you might infer a different meaning of the word. The related noun criticism shows a similar dichotomy.
The two senses of these words – one judgemental and fault-finding, the other neutral and evaluative – exist side by side in modern English, though the balance is uneven. With set phrases like literary criticism and film criticism, the analytical sense is a given. But more often the word is used negatively (He can’t take criticism), and the same goes for criticise.
When we express an opinion, we usually want to avoid giving offence – and when we offer criticism, the chances of doing so are considerable. So language has many strategies for being polite. For example, we may soften our criticism by using an expression like In all fairness, or Don’t get me wrong. Macmillan Dictionary has a list of these phrases, along with advice on how to use them.
The phrase constructive criticism implies that criticism can be destructive, and by using it we signal the intention to help, not to be simply disparaging. There’s also the neutral word critique. To revisit our earlier example, if I think you might misinterpret my offer to criticise your text, I could offer instead to critique it – this leaves no possibility of misunderstanding, though it does run the risk of sounding a little pretentious.
Critique probably grew in popularity as a result of criticise gaining pejorative connotations. Usage expert Kenneth G. Wilson wrote that it ‘appears to have become modish at least partly because it seems not to be loaded pejoratively as criticize is’. Merriam-Webster’s Dictionary of English Usage agrees that the negative overtones of criticise probably prompts many people to use critique instead.
Some critics find critique – noun or verb – superfluous or unworthy. Fowler hoped it would die out. R.L. Trask, in Mind the Gaffe, advised writers to use criticise instead, because ‘very many readers will object to the use of critique as a verb’. So we can add verbing to the list of reasons the word causes grief, despite critique being used as a verb in English since the 1700s.
Returning to criticism, Robert Burchfield said its two senses ‘coexist without risk of ambiguity or cross-infection’. But I think ambiguity is sometimes possible. I wouldn’t recommend using critique just because it sounds fancier than criticise or criticism, but if there’s a chance of readers misinterpreting or being uncertain of your meaning, critique may be the better option, or another synonym such as evaluate, appraise, analyse, or their associated nouns.

Another fine example is attitude, which in isolation mostly means ‘bad attitude’, though it was originally neutral. Much older is the similar transition of luck and fortune to the positive side, unless specifically qualified: hence the country-music line “If it weren’t for bad luck, I’d have no luck at all.”
Thanks, John – those are good examples of the same sort of thing. Luck in isolation can be negative, too, as in “Just my luck” (though that might be better considered simple irony). The C&W line reminds of the saying “No news is good news”, but it isn’t quite analogous.