The driving force is the same for the case of just mixing the reactants. By separating the two half reactions, the electrons must travel through the wire and we can use the electrical energy. In the sketch above the electrical energy powers a light bulb. When we first set up this electrochemical cell and complete the circuit, we could measure a voltage through the wire. As the reaction proceeds, the system approaches equilibrium and the voltage would eventually go to zero.
Batteries are electrochemical cells that convert chemical energy to electrical energy.
The reactants in batteries are at non-equilibrium concentrations. As you use them, the reactants form products to approach equilibrium and the voltage drops until the battery is no longer usable.