Elicit is a verb and it means to obtain. Usually it is used when you are trying describe drawing out information or to extract or obtain information. It also means to deduce or to construe.

For example:

Fog always seems to elicit strong feelings of melancholy.

The teacher asked a question to try to elicit information from the students.

Illicit is an adjective which means illegal or contrary to accepted norms or morality

For example:

We have been told to expect a purge on illicit filing-sharing websites

The act seeks to prevent the illicit trafficking of narcotics.

Both words sound very similar, but have completely different meanings!!! Using the wrong word in your sentence will completely change the meaning . . . and probably confuse your reader.