This manuscript presents an extensive comparative analysis between a staggered and a monolithic solution technique for the bidomain equations of cardiac electrophysiology in the context of finite element method. For the monolithic solution technique, we adopt the work of Dal et al. (CMBBE 15: 645-656, 2012), where the parabolic and elliptic partial differential equations (PDEs) of the bidomain model are solved simultaneously and all rate equations are treated with an implicit time integration scheme. For the staggered scheme, however, we simply suggest a decoupled solution of the parabolic and elliptic PDE while keeping other aspects of the monolithic algorithm the same, such as utilization of the implicit time integration scheme, coupling of ordinary differential equations, which describe the state variables of cardiac electrophysiology, to the parabolic PDE and consideration of the transmembrane potential and the extracellular potential as degrees of freedom. Both solution algorithms are applied to several problems where we simulate regular planar wave propagations, scroll waves and externally applied electrical fields for different spatial and temporal resolution and material parameters. The comparison between the solution schemes is performed in terms of accuracy, efficiency and stability. We reveal that the suggested staggered solution scheme yields almost identical results with the monolithic scheme for physiological conductivity parameters and small time increments. Besides, the staggered scheme provides an enormous gain in computational efficiency, particularly, as the number of degrees of freedom is increased and we do not encounter any stability issue arising from the decoupled solution of the PDEs. Therefore, we conclude that the suggested fully implicit staggered solution algorithm has a tremendous application potential for the bidomain equations of cardiac electrophysiology.