注重 means lay more emphsis on 关注 means care about / pay/attach more attention to

注重 means "think something important". 关注 means "focus on". E.g. 他很注重办事效率的提高。 He think it important to increase the efficiency of doing things. 他一直以来都在关注中国教育的发展。 He has been focusing on the development of Chinese Education.