Learn English – the difference between “maybe” and “may be”

Under the entry for laughable in Merriam-Webster's Eleventh Collegiate Dictionary (2003), a note on synonyms has this side-by-side comparison of ludicrous and ridiculous:

LUDICROUS suggests absurdity that excites both laughter and scorn {a thriller with a ludicrous plot}. RIDICULOUS suggests extreme absurdity, foolishness, or contemptibility {a ridiculous display of anger}.

So the distinction between ludicrous and ridiculous, MW seems to assert, comes down to the difference between absurdity that may provoke scorn and absurdity that may provoke contempt, within a larger context in which both words involve "provoking laughter or mirth." In my experience, ludicrous tends to be used most often to describe a situation that has elements of farcical incongruity, whereas ridiculous is a more common, all-purpose term whose meaning amounts to "not to be taken seriously."

As the following Ngram chart for ludicrous (red line) versus ridiculous (blue line) for the period 1700–2005 indicates, the frequency of use of both terms in Google Books' database of published works has decreased slightly but fairly steadily for most of the past century, and the difference in frequency between the two words has remained remarkably consistent:

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.