We already have a weather tag, and it already has climate as a synonym.
It makes sense for them to be synonyms because the questions often don't clearly differentiate, which is fine.
But it can look a little odd when a question is clearly talking only about climate but the tag says just "weather".
I think we would be better off including both terms in the tag. Let's have a little vote for which way to do it would be best... just because we can (-: