The basics of an atmosphere are pretty simple to imagine. You have a ball of gas surrounding a planet. It gets warmed by the Sun on one side and expands, cooling on the nightside. Chemistry occurs due to the heat of the Sun and the swirling of the atmosphere. Warm atmosphere is convected away.
Getting into the nitty gritty can cause a lot of different scenarios to come out – organic life can end up promoting oxygen and other chemicals through various cycles, as on Earth. Volcanic activity can pump out lots of sulphuric compounds, as on Venus, both altering the chemistry. The effects of sunlight vary according to wavelength, with deadly short wave radiation absorbed high in the atmosphere along with short wave stuff and only certain wavelengths like visible light allowed through to the ground, altering both temperature and composition. The amount of sunlight also varies according to the season at a particular point. The ground itself also scrapes along the bottom of the atmosphere and churns it. Particle precipitation from the aurorae also heats up the local atmosphere.
With all these actions, it is easy to imagine that the gas giants, free from vulcanism and ground, are easy systems to deal with. However, the hydrogen rich chemistry they have varies from that of the Earth, and in the case of Saturn, sunlight doesn’t quite hit the surface as evenly as expected. The rings block a bit out and also reflect a bit.
The effect of ring shadowing has been the subject of many a paper and is discussed in this recent webpage from the Cassini Equinox mission. Effects such as winds driving warm air into the cooler ring shadowed region and the chemistry of cooler gas amid warmer gas are spoken about as well as the changes with the seasons on Saturn, as observed by Cassini, which has been around long enough to see a warm southern hemisphere give way to a warm northern hemisphere.