Why do two car batteries with identical voltage readings have completely different car-starting ability?

To properly check any battery, whether its a car battery or a flashlight battery, it should be done under load. With no load, most batteries will show a normal terminal voltage. As batteries age, they develop an increasing internal resistance which causes the voltage to drop under load while still remaining high when unloaded. Check your battery with the engine off, both unloaded and with the headlights on; there should only be a small difference. Do this now with your new battery to learn how a good battery behaves.