You may not be making any mistake at all if you assume health science and life science are the same since both deal with the study of living organisms. However, health science is seen mostly as a subset or a branch of life science, meaning life science would be a broader field of learning compared to health science.
Life science refers to the study of anything about living organisms. This includes the study of everything that pertains to humans, animals, and plants. Life sciences strive to learn about all the functions of human beings, animals, and plants.
It also encompasses the study of some non-applied courses such as organic chemistry, zoology, biochemistry, physiology, botany, etc. On the other hand, health science is a field restricted to the study of human beings, animals, and their functions. Health science is the study of anything that pertains to the health care of human beings.