hardening laws controlling the material behavior after yielding somehow.

in isotropic one, the yield surface is expanding but in the kinematic one it is translating without any expansion.

read the Computational plasticity book by Dunne.

I can add that for monotonic loading, i.e. no unloading or reloading, you will not find much difference for choosing any of the models.

If you deform a solid plastically, then unload it, and then try to re-load it so as to induce further plastic flow, its yield point/elastic limit increases i.e. the material resistance to plastic deformation will have increased. There are, two simple hardening rules for modelling strain hardening in a finite element material: Isotropic and Kinematic. Isotropic hardening rule states that the yield surface expands proportionally in all directions when yield stress is exceeded. Kinematic hardening rule states that the yield surface does not exceed, but translates in the direction of the stress rising and stays in the same area and shape.