길 is broadly used to mean a way or road, while 거리 has a much narrower usage to mean a street or boulevard.
The two are usually not interchangeable since 길 has so much wider usage.
- 거리는 가로등 불빛으로 환했다 => The street was brightly lit by the street lights.
- 토요일 거리엔 행인들이 넘쳤다 => The Saturday street was overflowing with passers-by.
Different senses of 길:
- 서울로 가는 길 => The road to Seoul. (road)
- 여기서 부산까진 먼 길이다 => It's a long way from here to Busan (journey)
- 그 친구를 찾을 길이 없다 => There's no way to find him (way, method)
- 뜻이 있는 곳에 길이 있다 => Where there is a will, there is a way (means, solution)
- 눈길 (eyes fixing on something; sight line), 손길(touch/movement of a hand), 발길(path/movement of the feet)