How does the amount of mass in an atmosphere affect the seasonal changes and dynamics of the planet's climate?
Seasons are determined by the degree of tilt of the planet's axis, not the mass of the atmosphere.  Mars has an axial tilt slightly larger than Earth, so Mars' seasons are slightly more pronounced than Earth's.  For a planet like Venus, which has virtually no rotation and an extreme axial tilt, "seasons" are also affected by the rotation/revolution ratio.  

If you are interested in the climate differences between separate planets, mass of the atmosphere does affect overall global temperature (e.g. more atmospheric mass = more heat trapping) but the mass itself is not the main driver of the dynamic changes of climate within an individual atmosphere.  Really Mars and Venus don't have as much "short term" climate dynamics like Earth, because they don't have ocean/land dynamics.  Though, they definitely have seasons. Earth's "short term" climate dynamics are driven by the heterogeneous distribution of land, ocean, and (to a smaller extent) clouds.  

All the planets do share similar long term drivers of climate dynamics, which on Earth are collectively known as the Malenkovitch cycles.  The Malenkovich cycles are on the order of 10s of thousands of years and include changes in axial tilt, axial precession, and orbital eccentricity.  Some would also argue that long term gravitational changes in orbit parameters due to trojan asteroids and large planets like Jupiter also affect the Malenkovitch cycles.  These cycles shift the timing, intensity, and spatial distribution of solar input to the planet.