It's a bit hard to explain but I'll give it a try!
세상 is more broad but is used more metaphorically, as in this realm that people live upon. Examples: 세상에서 제일 좋아하는 사람 (my favorite person in the world) 세상을 떠나다 (to pass away)
세계 is more physical, like this geographical planet or the international world. Examples: 세계 지도 세계 뉴스 세계적인 화가
Note: 세계 has some other interesting uses that are also metaphorical, but very intuitive for English speakers!
세계 can mean world as in a way of thinking. 아이들의 세계 (the world of children) 나만 알고있는 세계 (the world that only I know)
세계 can also mean world as in a field of study/interest. 동물의 세계 (the world of animals)