I think ‘맡다’ and ‘담당하다’ are similar to ‘take charge of’ or ‘take care of’ or ‘handle’ or ‘manage’ in English. ‘부담하다’ is like paying for something. For example, 요리는 제가 맡을게요. Or 요리는 제가 담당할게요. I will take care of cooking. 이 레스토랑에서 제 남자친구와 먹은 저녁은 제가 부담했어요. I paid for dinner with my boyfriend at this restaurant.