mystery applies to what cannot be fully understood by reason or less strictly to whatever resists or defies explanation.
the mystery of the stone monoliths
problem applies to a question or difficulty calling for a solution or causing concern.
problems created by high technology
enigma applies to utterance or behavior that is very difficult to interpret.
his suicide remains an enigma
riddle suggests an enigma or problem involving paradox or apparent contradiction.
the riddle of the reclusive pop star
puzzle applies to an enigma or problem that challenges ingenuity for its solution.
the thief's motives were a puzzle for the police
Word History
Etymology
Noun (1)
Middle English mysterie "hidden religious truth, rite or event with religious significance, hidden meaning," borrowed from Anglo-French & Late Latin; Anglo-French misterie, borrowed from Late Latin mystērium "hidden religious truth revealed by God, religious rite, Eucharist," going back to Latin (in plural mystēria), "secret religious rites, things not to be divulged," borrowed from Greek mystḗrion, plural mystḗria "religious rite to which only initiates may be admitted, secret" (in the New Testament, "religious truth revealed by God"), from mys-, base of mýstēs "person initiated (into a religious cult)" + -tērion, suffix in words denoting instruments, place and ceremonies (derivative of -tēr, agent suffix) — more at mystic entry 1
Noun (2)
Middle English mysterie "ministry, office, craft," borrowed from Medieval Latin misterium "duty, office, occupation, trade," in part Latinization of Anglo-French mester, mister (continental Old French mestier) "function, duty, pursuit, trade, craft, guild" (going back to Late Latin mīsterium, variant of Latin ministerium "office of a servant or attendant, duty, support"), in part borrowed directly from Late Latin — more at ministry
Note: Medieval and Late Latin misterium are conventionally explained as a conflation of ministerium and Late Latin mystērium "hidden religious truth, religious rite, Eucharist" (see mystery entry 1), though a Late Latin form mīsterium (assuming manuscript spellings are early) would be a natural outcome of ministerium with loss of the pre-tonic vowel and nasal consonant.