The higher the power, the quicker the rate at which a battery can do work—this relationship shows how voltage and current are both important for working out what a battery is suitable for. Capacity = the power of the battery as a function of time, which is used to describe the length of time a battery will be able to power a device.
Current is expressed in Amps (A). It quantifies how many electrons are flowing per second. The capacity of a battery defines how much total energy is stored in each battery. The power output of a battery is how much energy a battery can give at a given time. This is a very important factor as it defines what you should use the battery for.
The capacity (at least to a first order) is the same in both cases. A battery's capacity is the energy stored, measured in amp hours, ergs, joules, or whatever unit you like. Watts are volts*amps or in your cases battery voltage times 1A, or battery voltage * 2A.
The charging current depends directly on the capacity of the battery, all other things being equal. When you read literature about batteries, you will come across C-rate. For example: "The battery was charged at 0.5C ." It's not temperature in Celsius, and it's not capacitance in Farads.
The energy stored in a battery, called the battery capacity, is measured in either watt-hours (Wh), kilowatt-hours (kWh), or ampere-hours (Ahr). The most common measure of battery capacity is Ah, defined as the number of hours for which a battery can provide a current equal to the discharge rate at the nominal voltage of the battery.
What is the difference between voltage, current, capacity and power? Electricity is commonly seen as the movement of electrons. Voltage is defined by how much energy each electron has as it moves. The voltage of a battery is defined by the elements in the positive and negative side (cathode and anode).