Meteorology is a study of short-term weather conditions, while climatology is a study of long-term weather, atmosphere and climate. It can be change in the climate and how different factors can affect the change. For example, climatology can tell how you can man-made factors are responsible for global warming and its impact on ecology.
Meteorology in simple terms relates more to weather forecast. For example, you get a weather forecast from your local meteorological department on a daily basis. Climatology on other hand is more comprehensive study of climate and weather effects and sometimes scientists monitor change in certain atmosphere conditions for years. For example, study of rising sea levels for years has now confirmed the scientists' view that global warming is happening in real.
Meteorology involves study of temperatures, heat, clouds and rains. In the study scientist observe short-term change in the air, its direction and then predict how the weather can turn out to be over the next day. These are basically short measurement of all contributing factors to short-term weather change. This is the reason that sometimes change in air direction happens faster than predicted by the scientists and prediction of an inevitable rain goes wrong.
Climatology involves study of atmosphere. For example, scientist will monitor that how temperatures levels in different part of world have changed over past 20 years, and then calculate how this change is going to continue over the next 20 years.