What's the point of having a different button for the battery? Why does the alternator turn on simultaneously with the battery? I need an easy and short explanation, if possible.
I will expand a bit on @Noah Krasser's answer with some situations where you would want one or the other. FWIW under-volt scenarios are typically far less of a problem than over-voltage scenarios.
Bad Voltage Regulator: Alternators vary their voltage output with RPM. To keep your system operating properly a voltage regulator is installed on the alternators direct output. This keeps your systems power supply at a constant voltage, typically at the cost of some power in the lower end. These things can and sometimes do fail, if they fail in such a way that the alternator is at a high RPM and putting out a voltage that could be potentially harmful to your avionics you are going to want the ability to cut it off while still drawing from your battery.
Battery Short: More often than not batteries are stored in the tail section of GA planes or potentially anywhere. If the cable chase leading to the battery develops a short you may want the ability to cut the battery circuit out while still having the ability to power you aircraft from the alternator.
Thermal Runaway: Batteries can runway on you and cause an over-volt issue. Again in flight you are going to want to be able to cut it from the system.
Short and concise answer:
Imagine there is a problem with the alternator. You have to shut it down. Now there is only one button to turn off both the alternator and the battery. You just want to turn off the alternator, but you would also lose your radios, the GPS, everything. Is that what you want?
In addition to the other answers, most modern alternators have stator and rotor coils. One generates the electricity, the other consumes current to generate a magnetic field. When driven by the engine, the current consumed is a small fraction of the current generated.
However, when you first power up your machine you really do not want the batteries delivering current to the alternator till after the engine is started and running or you will reduce the amount of battery power available and eventually drain the battery.
As such, having the ability to completely disconnect the alternator is a good thing.