There are some words that can take several different negative prefixes, and the different compound words have different meanings. Typically, one is the simple antonym and the others have narrower meanings. Non- typically means “something other than,” dis- “the opposite of, in a bad way,” a- “not having,” and anti- “opposed to,” but un- can be particularly flexible. Which words allow which prefixes is completely arbitrary. Some examples include:
Non-moral (not a moral issue), versus immoral (worthy of moral condemnation) versus amoral (a person who does not believe in right or wrong, an action taken without regard for morality, or something that lacks any moral dimension)
Uninterested (not interested) versus disinterested (having no conflict of interest). Watch out, though: some native speakers use these as synonyms, and others consider that an error
Nonfunctional (not functional) versus dysfunctional (functions in an awful or abusive way)—but note that dysfunction is a noun that means a failure to function, malfunction is either a noun or verb that means to function incorrectly, and there is no such word as *nonfunction or *malfunctional. (But there is malfunctioning.)
Nonsexual (content that is not sexual), versus asexual (an organism without sexual differentiation, or a person without a sex drive) versus antisexual (opposed to sex)
Unconcerted (not concerted, unplanned, not an organized effort) versus disconcerted (confused, perturbed)
Non-sensitive (not painful to touch) versus insensitive (not sensitive to other people’s feelings, callous)
Sometimes the base word and a compound that reverses it have evolved so much that they no longer are even opposites. The words ingenuous, disingenuous and ingenue have diverged to the point that they no longer have anything to do with each other. They all derive from the same word meaning “honest, plain-speaking, high-minded” (or literally, possessing “inborn” virtue), which came to mean naïve and honest to a fault, but ingenuous got mixed up with ingenious, an ingenue is an innocent young woman, and disingenuous means pretending not to understand or know things that you really do. An example is when antisemites disingenuously pretend that the word antisemitic means something other than hating Jews and act like it refers to the Semitic language group. The people who say that know better and are trolling.
That was also an example of how in- sometimes means the-opposite-of, but often is a fossilized Latin preposition instead. An ingenious person is clever and inventive, not the opposite of a genius.
Similarly, unbarred means not blocked or prohibited, but disbarred comes from a different root and means that someone was banned from the legal profession for unethical conduct. However, an unfrocked Catholic is a (rarely-used word for a) defrocked priest, not a layperson, and unrobed is a rarely-used synonym for disrobed.
In front of some ethnonyms, such as American, Christian, Muslim or so on, you can add negative prefixes with very different connotations. Non-Jewish is a neutral term meaning “not Jewish,” Anti-Muslim is a disparaging term meaning “bigoted against Muslims,” and un-Christian means “hypocritically antithetical to the values of Christianity.” The exact connotations vary.
Or to get really obscure, anti-establishment means trying to topple the powers that be, non-establishment means holding different views, but disestablishment has a very narrow meaning: abolishing what the U.S. Constitution calls a “law regarding an establishment of religion,” and revoking the special legal status of a state religion. There’s even a word antidisestablishmentarianism, the movement of people opposed to disestablishing the state religion, although most appearances of it are in lists of the longest English words.