The basic concept you need to know is that a battery tries to keep its voltage constant, it is what they are designed for.
When a load is connected to a battery, it is the nature of this potential (voltage) to deliver current to this load. The load is basically a way for charge to travel from high to low potential (one terminal of the battery to the other).
Since accumulation of charge is what creates the potential in the first place, if they find a way to go from one terminal to the other externally to the battery, voltage would decrease, so it is up to the battery's chemistry to internally force them through and elevate them back to the source terminal so that voltage can be maintained.
If the current being allowed to flow is too much for the battery to handle, voltage cannot be maintained and will drop. Hence the higher current capacity the battery has, the better it is at maintaining that voltage from dropping without getting internally damaged.
You see now that batteries can have the same voltage, but their current is a capacity rating, not something they are outputting all the time.