Both of these words appeared on my N3 flashcards, and they both can translate to cost or expense according to dictionaries. Example sentences had a tendency to translate 費用 as "cost" and 支出 as "expense", but I can't tell if that means anything significant or not.
In English, "cost" and "expense" are typically interchangeable, except in accounting where they mean separate but specific things. Is this true in Japanese with these words too? If not, what's the difference between them?
費用: the money for obtaining something—no matter who pay. Thus you can translate it as "cost".
支出: the money you spend or use—for whatever purpose. I think a better translation is "expenditure".
What tax officers find through investigation is 不明な支出. What an hotel charges you for nothing is 不明な費用.
I think the difference is exactly as you wrote. 支出 means simply a "pay out" or "expense", whereas 費用 can be used to mean "expense" but also the "cost" or "fee/price" for something. In certain contexts they can be used interchangeably but the meanings are different.
