absolutely, definitely

(ぜったい) Used by the person who is DOING the favor. “I’ll definitely do it!”
(ぜひ) This is used by the person ASKING for a favor, or by the person GIVING an order. “Please do it at all costs!”
However, these two expressions are not a set. That is to say, it’s not like ‘please’ and ‘thank you.’ If someone hits you with a 是非, you don’t have to respond with a 絶対 of your own.