I'm back home now, so I can pull my old (huge) Lewis and Short off the shelf. Here are the English equivalents it gives for "vates":
Original: a foreteller, seer, soothsayer, prophet
By transference: a poet, poetess
Post-Augustan: an oracle (i.e. a teacher, master, authority in any art or profession)
Even in 1879, they considered "bard" too archaic as a translation. :)
From Proto-Indo-European weh₂t-i- (“seer”), from weh₂t- (“to be excited”). (Wiktionary)
The English vates, despite being borrowed from the Latin form, is generally used about ancient Celtic seers rather than Roman ones. As far as vates (“celtic seer”) and bard (“celtic poet and singer”) are used in English, they tend to be somewhat overlapping.
The Latin vates is either a cognate with Proto-Celtic “*wātis” (especially if we suggest a common Italo-Celtic branch, as some do), or it is borrowed directly from Celtic. In any case, pagan Rome had a vates institution of its own... something like “seers” and “soothsayers”, or even “oracles”, residing on the Hill if of the Vates (Vatican Hill). The word ”vates” itself fell out of fashion, but Virgil revived it from old texts. Virgil is the Man who is the sole measurement of what is Classical Latin.
There are many cognates to be found in different Indo-European branches; most of them are more or less obsolete – including English adjective wode, (”mad, possessed, rabid”), Old English substantive wōþ (”sound, voice, noise, cry, song, poetry”), Irish fáidh (“seer, prophet”), and Sanskrit: वात (vā́ta, “attacked, assailed, injured, hurt”).
Beside vates, the only one of the cognates I’ve learned from Duo the Owl so far, is the German adjective wütend (”angry”).