They're all basically synonyms. They mean, essentially, "to free from blame". However, they're used in slightly different contexts. "Acquit" is used in court, specifically to refer to the process by which a judge or jury removes official charges of a crime against a defendant. "Absolve" is used more in a religious sense, where a priest would "absolve someone of their sin" or something along those lines, or far less commonly in a general civil sense to remove a person of an obligation (e.g. "I absolve you from your promise to cut my grass"). And "exculpate" simply isn't used anywhere that I'm aware of, except maybe in court briefings.

You can add the word "exonerate" to your list, which is still another word meaning "to free from blame." Unlike "acquit," which is the best word to use after a court or jury has found the defendant not guilty after a trial, "exonerate" can be used even when there has been no trial.