This is a prehistory question. There are reconstructible words for both in Proto-Indo-European, which is about 6,000 or about 8,000 years ago (depending on which of the major theories you subscribe to), so they're both far older than any source we have that mentions either one. (There's also a word that probably means mead, but night just refer to honey.) Evidence from molecular archeology further supports both being older than anything we recognize as a major civilization.

There's a conjecture - but that's all it is - that, because mass production of beer rewrites agriculture but significant amounts of fruit wines can be made by gathering wild produce, wine must be older. I find that reasonable, but even its most ardent adherents admit it's not proven. The notion that mead may well be older than either is popular among those who make it, eg appearing in The Compleat Meadmaker by Ken Schramm, but this fact merely means it's entered brewing folklore now, not that it's the truth.