‘Elicit’ and ‘illicit’ are homophones, but they have different spellings and meanings and one must be careful about using them.
On the one hand, ‘elicit’ means to draw something out of someone, by coaxing or pleading or urging. For example, the sentence - His father was unable to elicit a response from him on the matter means that the father could not extract the information that he desired from his son.
On the other hand, ‘illicit’ means illegal, or forbidden. Examples: Theirs was an illicit relationship because their parents were opposed to inter-caste marriages, or He was thrown into prison for smuggling illicit weapons into the country.