The Cassini mission has opened many science questions related to Saturn's atmospheric dynamics: jet-streams (including the hexagonal-shaped northern polar jet), giant convective storms, stratospheric equatorial oscillations, interhemispheric transport. This inspired our group to develop a Global Climate Model for Saturn that couples a massively-parallel icosahedral-grid hydrodynamical solver with detailed physical parameterizations (notably correlated-k radiative transfer) for Saturn's troposphere and stratosphere. Multiple test simulations led us to determine the best modeling settings to run simulations in which (amongst other requirements) the conservation of angular momentum is ensured with good accuracy. Multi-year simulations carried out with a spatial resolution of 1/2° in latitude/longitude allowed us to analyze the mechanisms accounting for jet acceleration (and possible latitudinal migration) in the troposphere and the stratosphere of Saturn, the formation of large-scale vortices and waves, and the occurrence of a zonostrophic regime. The perspectives for improvements of the reference Saturn simulations will be reviewed. We will also present simulations carried out for Jupiter (presently observed by the Juno mission) with the exact same modeling platform, to discuss comparative dynamics between both gas giants.