To elicit is to give rise to, to draw out, or to evoke. The word only works as a verb. Illicit is an adjective meaning illegal or not approved by custom. The words are not quite homophones, but they sound similar enough to elicit occasional confusion.
Good incentives can elicit greater effort. [New York Times]
Psilocybin would be infused into their bloodstreams before a psychotherapy session, tailored to elicit positive memories. [Independent]
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 (dead link)]
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]