Elicit: verb, to draw out, to bring forth, to call forth or provoke.
Illicit: adjective, not legally permitted or authorized, disapproved of, as in an illicit affair or illicit drug traffic.
It is the story-teller’s task to elicit sympathy and a measure of understanding for those who lie outside the boundaries of State approval.
—Graham Greene Speech on receiving the Shakespeare Prize Awarded by the University of Hamburg, Germany, 1969
Grammar.com’s section on Problem Words discusses elicit and illicit. Click here for that discussion.