전혀(jeon-hyeo, jeonhyeo)
전혀 means that something does not exist at all or is not the case in any way, expressing strong absolute negation.
When to use
Strong absolute negation, Complete negation emphasis, Formal or written contexts (often), Absolute zero degree
The nuance
전혀 must be used with a negative expression (안, 못, 없다, etc.) and expresses absolute, strong negation.
별로(byeol-ro, byeolro)
별로 expresses mild negation or dissatisfaction, often indicating something is not quite up to expectations, mainly used in casual spoken language.
When to use
Mild negation, Expressing dissatisfaction, Casual spoken language, Indicating something is not satisfactory
The nuance
별로 is used with negative expressions and indicates relatively weak negation or dissatisfaction, natural in spoken language.