Learn English – the difference between “exhausted” and “enervated”

Skeptical means “having reservations”. Someone who is skeptical will not easily be convinced, will be hard to persuade.

The main meaning of cynical is “believing the worst of people”, or as NOAD says, “distrustful of human sincerity or integrity”. By extension, it has come to mean “doubting something will happen”, which can be somewhat close in meaning to skeptical in some cases, but not generally.

As an example, in the sentence “John is skeptical about the motorway extension”, you could replace skeptical with cynical without altering the meaning. In the sentence “John is skeptical about global warming”, you could not do this change. Likewise, in “John has a cynical attitude”, you could not use skeptical instead.

Something illegal is forbidden by law: "It is illegal for motocyclists [sic] not to wear a crash helmet."

If a thing is illicit it is done by someone who knows that it is disallowed by law but that under different circumstances it could be legal: "The crew were involved in the illicit import of brandy"

(it is basically legal to import brandy but not the way they did it).

This is backed up by things like illicit sexual relationship. It's not wrong to have a sexual relationship normally, but it could be "wrong" if you are married, or not yet married, etc. illicit trade is another such example. It's not wrong to trade, unless you do it in the wrong way.