Elicit is a verb meaning to give rise to or evoke. Illicit is an adjective meaning illegal or not approved by custom. The words are not quite homophones, but they’re similar enough in sound to engender occasional confusion.
Examples
In these examples, the words are mixed up:
Favorites like “Ever Death Metal Band Out of Denton,” “No Children,” and “This Year” brought the crowds to an impressively hard to illicit back-and-forth sway. [Daily Cougar]
… the case is strong with court approved wiretaps of the alleged drug deals, and 20 search warrants that resulted in seizing 200-thousand dollars in elicit drug proceeds … [WFMJ]
… underappreciated Aussie actor Noah Taylor adds droll humour and illicits sympathy as Oliver’s depressed dad … [Warrnambool Standard]
And these writers use elicit and illicit well:
Good incentives can elicit greater effort. [New York Times]
For the curious souls who fall prey to dark influence and illicit highs, it’s easy to get mixed up in a worrying subculture … [Irish Times]
Psilocybin would be infused into their bloodstreams before a psychotherapy session, tailored to elicit positive memories. [Independent]
The U.S. also wants Burma to open up to U.N. nuclear inspectors and sever illicit military ties with North Korea … [Sydney Morning Herald]

