There is a difference between energy density and power density. Energy density is how much charge is stored in the battery. Power density is how much electricity can be delivered per period of time at a point of charge.

A battery may have 2,000 milliwatts of charge, but can only deliver 500 milliwatts per hour; thus, it could run at full blast for ~4 hours or half at ~8 hours if the power density is not dependent on the charge. Batteries normally have a power curve, the amount of power you can draw depending on the charge in the battery. When you over draw the power on a battery, the tendency is for the voltage to quickly drop.

Cycling the battery and age both play in lowering the energy and power densities, but they are not one-to-one correlated. This is why you'll see people who say that their battery will last to 40% and die. The battery's voltage is used to gauge how much charge is left in the battery; however, at that level, the power density has dropped such that any significant draw on the battery causes the voltage to plummet and the phone shuts off.