Interaction between rubber and snow surfaces yields intricate challenges that have been a subject of scientific investigations. To enhance tire traction in winter circumstances, a thorough understanding of the effects that take place in the contact patch between the tire tread and snow surface is required. Describing the effects of individual tire tread block parts during sliding on snow, and milling is the major goal of this study. Recognizing the inherent complexities posed by these interactions, especially under varying applied velocities, this study aims to deepen the understanding of these characteristics. To achieve this, a dual methodology is employed: detailed numerical simulations and rigorous experimental validations. Considering snow behavior to exhibit brittleness and transition to large deformations under high strain rates, an advanced elastoplastic constitutive model at finite strains, enriched further with an implicit gradient damage enhancement, to replicate the snow's brittle behavior is used. The simulations offer insight into distinctive deformation patterns of rubber. Furthermore, the strain-softening behavior exhibited by snow, especially at the range of applied velocities, introduces additional aspects of complexity. The rubber model is subjected to a range of computational scenarios, in parallel, experimental studies, conducting precise material testing on rubber interacting with snow, are employed to validate the numerical simulations, ensuring the reliability and applicability of the developed models.