Elicit is a verb.
Meanings: bring out; draw out; obtain; extract.
Illicit is an adjective.
Meanings: against the law; illegal; banned; forbidden.
1. Despite repeated questioning, the police Inspector was unable to elicit a response from the thief about where exactly he had hidden the stolen goods.
2. Dealing in drugs is an illicit activity in all countries.
3. When the Professor was unable to elicit any response to his query from the students, he realised that they did not seem to have understood his explanation.