Question

*Please help!*
What produces the potential difference in a battery, and why do batteries with the same potential difference produce different currents?

Answer 1

okay, there is no such thing as an ideal battery. So a 12V battery is not going to actually produce 12V due to internal resistance.

This internal resistance is also what causes different currents.
Ohm's law is V=IR so I=V/R

Even if two batteries have the same voltage, different internal resistance will cause different currents.

Answer 2

Adding to what @roadjester provided, which accounts for variations "within" the same battery class (for example, car batteries). Between battery classes, such as between a car battery and a button battery, that is used to power watches and calculators, batteries can have the same potential. The difference is in their ability to produce current, which is to say that a car battery can provide more energy per unit time (i.e. more power) compared to a small button battery. This difference is simply due to amount of charge that is stored inside each battery.