Weather is what happens outside. The air, or atmosphere, around us behaves in different ways. It changes when it's hot or cold, and when it's wet or dry. It acts differently when it's calm or stormy, and clear or cloudy. The atmosphere reacts to everything from rain to sunshine.
Snow is weather.
Thunder and lightning are weather, too. Sometimes the atmosphere behaves violently, and sometimes it's peaceful and quiet. Either way, it's weather. Climate is the average weather in a location over a long period of time. A month without rain would be considered a dry climate.