Well, not exactly; it's a valid question- a new battery does make a big difference. It's true that the alternator pretty much runs everything once the engine is running. But you have to remember that the battery is also seen as an electrical load by the alternator.
As long as the charging voltage (typically 14.4v) is greater than the battery voltage (12v), that greater difference of potential keeps the current flowing to the battery. This is because the higher voltage is able to overcome the internal resistance of the battery. Thus, if the battery is old and doesn't charge as fast or as well anymore, the voltage difference is even greater and it becomes even more of a burden for the charging system, leaving less current to power everything else. That's the difference in performance that you're experiencing