Elicit involves the use of wit and logic to arrive at the truth by means of evoking feelings, responses and emotions. It is getting information from acute deduction.
"The FBI agent solved the case using elicit methods."
Illicit involves the unlawful breaking of social norms, customs and tradition to attain a goal, aim or objective. It is mostly illegal.
"Rumor has it that he got his money through illicit means."