Explain Why Two Words That Look Like Synonyms Are Not Interchangeable

Two words translate to the same English word in the dictionary, so learners assume they are interchangeable and get corrected by native speakers who cannot quite explain why. This gives the precise distinction in meaning or register, two example sentences for each word, and the one sentence where swapping them gives the game away.

Explain why [word A] and [word B] in [target language] are usually translated the same way into English but are not actually interchangeable. Give the precise distinction in meaning, register or context between them, provide two example sentences showing correct usage of each, and name one sentence where swapping them would sound wrong to a native speaker.