In pseudopotential lattice Boltzmann (LB) models for simulating multicomponent flows, interaction forces between the components of a mixture lead to phase separation and interfacial tension. At the macroscopic scale, such LB models solve an advection-diffusion equation for each component and the Navier-Stokes equations for the fluid mixture. In this paper, the computational efficiency of the LB method is compared with a finite volume (FV) solver for the same macroscopic-scale equations for a binary system in a two dimensional domain. The FV implementation replicates the phase separation of the LB model. Differences in the interfacial tension are due to truncation of the Taylor series expansion of the LB interaction force in the FV version. While the computations required to update the domain for each timestep can be completed faster with the FV approach, a smaller timestep is required to achieve stability, which negates the improvement in processing speed. The FV implementation, however, allows independent variation of model parameters, which is not possible in LB. For example, the viscosity can be changed without affecting interfacial tension or the extent of phase separation. Furthermore, it is possible to obtain low interfacial tensions without suppressing phase separation with the FV formulation. The significance of changing the diffusion rate of components on the deformation of a droplet in shear is also demonstrated. For three-dimensional simulations, the finite volume approach is expected to be faster than LB and would benefit from the demonstrated flexibility in specifying model parameters.