..But there are two subjects. If it said "there is neither tea nor coffee," then that would be correct.
Både is used with och as a conjunction, as you see here, to mean "both x and y."
Båda is used as a pronoun, and basically whenever you wouldn't want to say both without "both x and y," such as:
Två bilar i krock. Båda började brinna. (Two cars in crash, both caught fire.) Or
Jag skriver ut på båda sidorna av papperet. (I print on both sides of the paper.)
I don ‘t think you mean pronoun in your answer: maybe in the first (both caught fire), but not in the 2nd (both sides of the paper). There it is serving as an adjective, which is the lesson this question comes from.
This question hasn’tbeen Fully answered so far. We were taught ‘båda och.’ When is it used? And how is it different from ‘både’?