I'm a French student who currently learns English. But, one thing my teacher said intrigued me.

He said that we must use "Why?" when we want to know the reason as in "Why did she refuse?" (Answer: "Because...").

And "What ... for?" when we want to know the purpose as in "What did you send that letter for?" (Answer: "To...").

However, I've searched on the Internet to understand the nuance and almost everybody says they're both interchangeable. Wiktionary even says that "What ... for?" is a colloquial form of "Why?", and my french-english dictionary gives me the same translation for both, without any explanation. I'm a bit confused, now...
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Quoi -What
Porquoi - Why

You teacher is correct. They aren't interchangable. You could ask the similar questions but change these words and you will get very different answers.
Yes, but "What ... for?" is translated by "pourquoi" too in my dictionary. And it seems to me we could get the same kind of answer.

"What did she send that letter for?" - "To remind him of his promise".
"Why did she send that letter?" - "To remind him of his promise."

Wouldn't it mean the same thing?

"Why did she refuse?" - "Because it was a stupid idea."
"What did she refuse for?" - "Because it was a stupid idea."

Doesn't sound right to me.

I still don't understand the nuance.
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They are interchangeable in some cases, but I agree that 'what for' does carry a suggestion of purpose.

eg 'Why does water freeze below zero degrees?' is fine,
but ' What does water freeze below zero degrees for?' sounds odd.

Long story short, Wiktionary got it right--sort of. This basically defaults to the debate of ending sentences in prepositions. If you end a sentence in a preposition unnecessarily, the preposition shouldn't be there. (e.g., Where are you at? should just be Where are you?) Some say What for? (which ends in a preposition) is unnecessary because it means pretty much the exact same thing as Why?, and I tend to agree with that assertion. Additionally, Clive made a good point in that using What for? is sometimes more awkward than using Why?.

When push comes to shove, learning a foreign language is about communicating with native speakers. Since native speakers use What for? regularly, there's no reason you shouldn't. Being a bit of a grammarian, I personally don't use What for? myself, but the vast majority of English speakers do, so you don't have to worry about it. Ninety-nine percent of people don't even consider What for? to be an error, so if you use it, that's perfectly fine.

Just know that they basically mean the same thing. Whatever difference your teacher gives is negligible, if it be present at all.
I'm not a native speaker, so I'd be glad to get feedback on my thoughts about this.

It seems to me that when people use the "what for" construction, they're usually implying that they believe the action they're asking about is unnecessary or foolish or puzzling, that is, they have a negative attitude towards it (even if just slightly negative).

So for "What did she send that letter for?," the speaker is not only asking for an explanation, but also expressing some misgiving about the letter being sent. But for "Why did she send that letter?," depending on the tone of voice, the speaker may or may not have a negative attitude towards the action. He/She could be feeling neutral about it and is just curious about the other person's motivation.
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I absolutely agree with you.

I'm not a native speaker...
They are interchangeable only in some cases. Consider:Why is the girl crying? Because she is sad./Because she wants a computer. (reason/purpose)What is she crying for? Because she wants a new computer. (purpose).
Why = Pourquoi
What for = Pour quoi

Same in French and in English Emotion: wink
Meme chose en Francais comme en Anglais Emotion: wink
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