En quoi ma relation avec elle t'intéresse-t-elle ?
Pourquoi ma relation avec elle t'intéresse-t-elle ?
How does using "en quoi" here differ in meaning from a straightforward question with "pourquoi"?
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En quoi asks for a specific detail (what/how?) while pourquoi asks for a global motivation/explanation (why?).
En quoi : What aspect of my relationship with her interests you?
Pourquoi : Why does my relationship with her interest you?
Assuming these questions are asked in a jealousy context, I would also say the first question sounds more aggressive (i.e. it's none of your business) while the second one is more neutral (there is nothing to worry about).
Here are other contexts showing the difference between these expressions:
En quoi es-tu venu ici ? How did you get here? (by car/plane/bus/...)
Pourquoi es-tu venu ici ? Why did you get here? (because I wanted/needed to)
En quoi puis-je vous être utile ? How can I help you?
Pourquoi puis-je vous être utile ? (not idiomatic) Why can I help you?
En quoi vas-tu te déguiser ? What are you going to dress-up as?
Pourquoi vas-tu te déguiser ? Why will you disguise yourself?