There is a large variety of weather in the game. Various weather effects include snow, rain and even tornadoes.
Weather has two primary effects on the citizens in Banished; Warmth and Crops, both tied to temperature.
Low temperatures cause a drop in warmth of citizens, and at a certain point they will need to find a warm place to wait for a few moments to raise their temperature. If they are unable to find somewhere to warm up, they will be in danger of freezing, and will eventually die if the situation does not change. Different types of clothing can prevent cold for longer periods of time, enabling citizens to stay out in cold temperatures longer, making them more efficient.
Crops can be severely impacted by low temperatures. Weather is, to a degree, randomized, meaning that just because it is Early Summer or Late Winter does not guarantee a specific temperature. Indeed, temperatures can hover in the upper sixties well into the days of Late Autumn, and cold snaps below freezing can appear as late as Mid Spring. Because crops are sensitive to temperatures when they are planted and while growing, these unusual but very possible conditions can have big impacts on a year's harvest. If the temperature is too low when a crop is planted, the seedling will freeze, and will not continue growing, reducing overall yield come harvest time. Similarly, once the harvest is underway, if winter temperatures arrive early, plants will begin to die in the field before they have been gathered, preventing them from yielding any food.
Each crops reacts differently to specific conditions at specific times. Cabbage, for example, appears to thrive with early, warm temperatures that last into Mid Autumn. Squash, however, will tolerate even freezing temperatures at the end of Autumn, preserving a full harvest. Having multiple types of crops during a single season will ensure that a cold snap does not devastate your food stores.
A tornado can be seen in this 1 hour long gameplay video at 27:10.