Generalized Wasserstein distances allow us to quantitatively compare two continuous or atomic mass distributions with equal or different total masses. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the past few years, giving some insights into their physical meaning. After that, we explore their usage in the context of a sensitivity analysis of differential models for traffic flow. The quantification of the models' sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.