It's primarily a question of the letter v. If the word "vote" were to be equivalent to bhót (or something similar) I don't think I would have raised the question. Since the word as Ghaeilge is spelled vóta, though, I feel that it is more likely to come from a more recent borrowing; and there are plenty of words in the Duolingo course that suggest recent, comparable borrowings.
Scilling's answers sounded a little rude and pedantic, in my opinion, but please don't be discouraged; I'll attempt to answer your question. Guth is the native Irish word for "vote", but its meaning is broader and more ambiguous. Teanglann.ie defines it as voice; singing-voice; note, sound; utterance; expressed opinion; vote; and even "blame". I would say that vóta was probably added to the course, namely to the politics section, to eliminate the ambiguity and strictly mean "vote" when you want to be super specific.