Motion by mean curvature is a problem arising in multi-phase thermomechanics and pattern formation. The article presents a numerical comparison of two approaches to the dynamics of closed curves, namely sharp-interface description leading to a degenerate-diffusion equation of slow and fast diffusion types related to anisotropic curve shortening flow, and a diffusive-interface description in the form of a recently improved version of the phase-field model. Numerical scheme used for simulation of (isotropic) phase field equation is analyzed with regards on the convergence, and simultaneously existence, uniqueness and asymptotical behaviour if the diffusive interface becomes sharp. The presented computational results indicate a consistent relation of both approaches and demonstrate the behaviour of both models in different situations of curve dynamics.