Unless they pass very close or actually strike the Earth, Comets travel far out into the Solar System and so do not come close enough to Earth to affect its weather. The only way a comet would affect Earth (aside from a strike) would be if it possessed powerful gravity greater-or-equal to that of the Moon. A gravitational body passing close to Earth could, in theory, affect the tides just as the Moon does, and so affect ocean and weather patterns. However, this has not happened in recorded history.
If a comet were to strike the Earth, depending on size, it could throw up clouds of particulate matter into the atmosphere. This scenario is called an Impact Winter; this would increase reflectivity and allow less and less sunlight and heat to enter the atmosphere, causing the loss of heat over time. This scenario could potentially last for years, causing a moderate-to-severe Ice Age. Of course, depending on the size of the impact, billions of people worldwide would already have died.
Other than that, comets do not affect seasonal weather, rainfall, or any other weather patterns.