What is the difference between 专家 and 达人?
专家 is the formal word for 'expert'. 

达人 (literally 'achieved person') is first borrowed by Taiwanese from Japanese in no more than 30 years ago and gradually become known to mainland China. Some say the Japanese usage was derived from ancient Chinese, but I don't know if there is a definitive etymology for that. Anyways, it is not a formal word in Chinese (yet) and mostly used in entertainment topics.

Reference: http://baike.baidu.com/view/2453.htm