"Why doesn't he answer me?"
Translation:למה הוא לא עונֶה לי?
So if I understand what you're saying, the questions, "?למה זה קורה" and "?מדוע זה קורה", both translate as, "Why is this happening?", right? And the nuance is that the former is asking, "What purpose does it serve that this happening?" and the latter is asking, "What events led up to this happening?". Not literal translations, of course, but am I in the ballpark?
This response (by GamalNemeri) makes sense, esp. since למה can mean "in order that." And as radasgast wrote, מדוע is more formal. You might use מדוע in a conference paper, for instance. In an article on interrogative particles in the Encyclopedia of Hebrew Language and Linguistics (vol 2, pp. 316-18), Ruth Burstein refers to both as "interrogative adverbs of cause" and writes that מדוע has a "higher register," which is ironically an academic way of saying it's more formal; it tends to occur "in literature, in interviews on economics, and in textbooks" (p. 318). As GamalNemeri noted, so also Burstein says למה can be used to ask about purposes and aims.