1. How come one person usually pays for everybody in Korea?
2. Why does one person usually pay for everybody in Korea?
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Are both the same and grammatical? What is the usage difference between the two sentences?
Yes, both sentences are correct ... and they mean the same thing.
The only real difference is based on the use of "How come" or "Why" as the interrogative. "How come" results in the rest of the question being in normal English word order, but "Why" results in the somewhat reversed word order that is normal in English questions.