If aging is getting older and dying is getting closer to death, then we are aging and dying every day. This is not, however, what people usually mean when they talk about aging and dying as things they fear or worry over. What they mean by "aging" is showing signs of being old. In this sense, you are already old when you are aging; and in the process of aging, you are getting more and more signs of being old, and those signs are getting more and more obvious. What they mean by "dying" is approaching death in such a way that death is imminent and caused by some kind of illness. The final stage of aging overlaps dying, for the end of aging is not just getting very old, but death. (But the final stage of dying does not have to overlap aging, for a person can die young.)
I do not fear death, for I see my death as nothing but the non-existence of me (again). But I fear both aging and dying because I fear physical and mental decline, and fear the pain that is likely to accompany the illness that will be the cause of my death. Because of this fear, I have this irrational inclination to believe that I will not live to an old age. This is, I think, a weak form of wishful thinking.