Mystery, problem, enigma, riddle, puzzle, conundrum are comparable when they denote something which baffles or perplexes and challenges one’s power to solve it.
Mystery may, especially in theological use, imply the thing’s incapacity for comprehension by the human reason, but it is also applicable to any of the facts of the world about us which defy all attempts to explain their cause or nature.
The term is also used to denote something which is guarded by secrecy or is in itself or by design mystifying while at the same time so intriguing as to compel speculation.
Problem applies not only to a perplexing question that demands a solution, but also to a person or situation that causes perplexity or puts one in a predicament.
Enigma applies to whatever hides its meaning under obscure or ambiguous allusions so that one can only guess at its significance. The term can be extended to whatever is inscrutable or beyond the range of unaided understanding.
Riddle applies to an enigma involving paradoxical or contradictory statements and definitely proposed to be guessed. The term can be extended to any problem which is difficult because of its inner contradictions.
Puzzle applies to a problem or enigma which tests one’s ingenuity or skill in solution or which is peculiarly baffling.
Conundrum specifically applies to a riddle phrased as a question the answer to which involves a paradox or a pun or an equivocal use of words. The term can be extended to unsolved or unsolvable problems which provoke speculation rather than serious attempts at solution.