All of the three words mean ‘why’. 为什么 is the most general word, and it can be used in any contexts. 为何 sounds a little quaint, so it is used more in written and formal language. 为啥 is colloquial and is actually from northern dialects.
The meaning of these three words is basically same, but “为何” is more formal, and is often used in written expression, we seldom use it in spoken language. “为什么” can be used both in spoken language and written expression. “为啥” comes from a dialect, and it is informal.We hardly use this word except when we are chatting with our friends.
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