I would say "소중하다" is used when talking about something personal (ex. 이건 저에게 소중한거에요(=this is precious thing to me), 제겐 소중한 거니까 조심히 옮겨주세요 (=please carry it carefully because it is precious thing to me)). And "중요하다" is often used when talking about business or something not personal (ex. 이건 국제적으로 중요한 문제입니다(=this is internationally important matter), 이건 그가 그곳에 있었다는 중요한 증거입니다 (this is important evidence which proves that he was there)).