We study the classical and the quantum dynamics of a particle in a double square-well potential driven by a sinusoidal external force. Classically, using both the renormalization technique and a computer simulation, we compute the critical amplitude of the driving force at which the destruction of the KAM tori lying between two neighboring resonances occurs. The critical amplitude is seen to take on a higher value when the double well potential is symmetric with respect to its center than when it is not. Similarly, our quantum calculations performed on the same system indicate that the probability distribution tends to spread less and the entropy tends to be lower when the potential is symmetric. Investigation of the structure of the resonances in the classical phase space and the corresponding quantum considerations yield a clear physical picture which establishes the relationship between the symmetry of the potential and the dynamics subjected to that potential.