This is the 194th most frequent Korean word.
달리
"달리" means "differently" or "in a different way" in English.
In this sentence, '달리' means 'unlike' and is used to indicate a contrast between the mentioned subject ('그') and others ('다른 아이들').
다른 아이들과 달리 그는 공부에 관심이 많다.
Unlike other children, he is very interested in studying.
Here, '달리' serves as an adverb modifying the verb '느긋하게' ('leisurely'), which is together used to describe how the walking was carried out.
그는 산길을 달리 느긋하게 걸었다.
He walked leisurely along the mountain path.
In this context, '달리' again denotes a contrast, but this time between her current state and a state from before ('예전').