The difference between weather and climate is the measurement of time we associate with them; weather is related to short-term changes in the atmosphere while climate is the average weather over a long period of time. You could also think of the difference between the two as, "weather is your mood, climate is your personality."
Weather is the balancing of the atmosphere. We associate weather with sunshine, rain, cloud cover, winds, hail, snow, sleet, freezing rain, flooding, blizzards, ice storms, thunderstorms, steady rains from a cold front or warm front, excessive heat, heat waves and more.
Then climate is the way of describing weather patterns over a long period of time. Climate is a way for scientists to visualize the average weather over periods of ten or fifty years. Studying climate is important because it affects many people around the world. Many types of ecosystems could be altered permanently due to rising global temperatures which will affect plants, humans and other animals. "Rising global temperatures are expected to raise sea levels, and change precipitation and other local climate conditions. Changing regional climate could alter forests, crop yields, and water supplies."
Climate change does make the weather more severe, such as hurricanes, flooding, and droughts: the amount of water vapor that the atmosphere can hold increases with temperature. Just because colder months still happen, does not mean that global warming is not presently happening. Though just because there are a few more warm days in November does not completely prove there is global warming either.
This is not to say that climate change is not real because global warming is in fact happening. The planet has been experiencing more severe weather patterns and loss of ice at the poles due to an overall higher average temperature that is only increasing. Greenhouse gas emissions are trapping heat in the earth's atmosphere causing this detrimental rise in temperatures.