Extramarital affairs are against the law in many countries and 17 US states. “Illicit affair” is potentially a holdover from when it was illegal more places, not just a conflating of against the rules with illegality.

https://en.wikipedia.org/wiki/Adultery_laws

"Licit" in English usage also carries the weight of its connotation in church ("canon") law, wherein it can refer to acts that are considered not only to have occurred spiritually ("valid") according to the law, but have been done in conformity with the process laid out in the law.

For example, canon law asserts that certain sacraments such as marriage or baptism can be conducted validly but illicitly, such that the marriage is still considered to have occurred, but it wasn't done the "right way." In contrast, there are some things that, if done illicitly, aren't valid either. (In this case I think it's because there's a notion that there's a supernatural/spiritual aspect to certain acts that transcends whether they're done licitly, whereas certain other acts are purely creations of human law and therefore if not done licitly are void ab initio.)