They means similar. The most usually means 가장, 제일 and 최고, and the best means 최고 and 제일 좋은. The most difficult question=가장/제일/ 최고 어려운 문제. The best education= 최고의/ 제일좋은 교육, the best day= 최고의/ 제일 좋은 날.

the most, the best. In Korean, we have lots of expression up to the situation. But, your example can be translated to the below. The most good manner one.(The best manner one) 가장 좋은 매너를 가진 사람