Despite their similar pronunciations and spellings, these are in fact completely different words with very different uses and meanings. The word elicit is from Latin elicere 'to draw forth', itself formed from ex- 'out' + lacere 'to entice or deceive'.
Illicit was introduced into English in the 17th century from French illicite, which is from the Latin illicitus. This Latin word was formed by adding the negative prefix in- to the past participle of licere 'to be allowed', so that illicitus means 'not allowed'.
Given these differences, the two words should be easily distinguished: elicit is a verb meaning 'draw forth, evoke' and illicit is an adjective meaning 'not allowed, forbidden':
The comment elicited a strong response
The government is cracking down on the use of illicit drugs
Despite this simple distinction, even reputable publications can confuse the two words. In 2010 the British Medical Journal included an editorial on 'Evidence based policy for elicit drugs'! The opposite mistake crept past The Guardian editorial team in an article published in 2002, which contained the phrase 'illicit a response', prompting a correction in 'near homophone corner'.