How does the build up of plaque in someone with atherosclerosis cause the hardening of the arteries?

Not the same things. Plaque buildup (atherosclerosis) in the arteries can make them less flexible, but the old term "hardening of the arteries" (arteriosclerosis) is usually attributed to a decrease in flexibility of the vessel wall even without any plaque present. It is more related to the collagen/structure of the vessel wall themselves. The utility of this term is waning as medicine advances.