I think both mean "to boil" but what is the difference between them? Both seem to be used in the context of cooking.
'끓다' is about water. Like, '물을 끓이다'((I) Boil the water.)/'물이 끓다'(Water is boiling).
'삶다' is about something inside water. Like, '감자를 삶다'((I) Boil the potato) or '감자가 삶아지다'(Potato being boiled).
So, sentence like this '물 끓이고 거기에 라면 좀 삶아줘'(Boil water and boil some ramen there) is correct, but '물 삶고 거기에 라면 좀 끓여줘' is wrong. (to be honest, Even though it is wrong, '라면을 끓이다' is large-used idiom.)
In short, '끓이다' is 'to boil liquid', and '삶다' is 'to boil something inside water'.