Please explain the main differences between Climatology, Meteorology and Atmospheric Science.
Atmospheric science is the generic study of the atmosphere. This includes climatology, air quality, and meteorology. It is often all-encompassing.
Climatology- the study of climate, the atmosphere over an extended period of time. This can usually be seen from either a statistical or physical perspective. Usually, the statistical approach looks at climate extrema. Physical climatology studies the processes that drive climate, and may be used as reconstruction, or prediction. Some of the processes that meteorologists may neglect, such as aerosols and long term trends in oceanography, are very relevant in climatology. Climatology is also a subset of atmospheric science.
Meteorology is the study of weather, especially in forecasting. Often, meteorologists interface with the public, via television, forensic meteorology, produce forecasts etc. In regards to research, meteorologists often study improvements to atmospheric models, or other aspects to forecasting. Some research meteorologists study mechanisms behind certain weather phenomena, in order to improve understanding, say of severe weather, atmospheric chemistry, turbulence, etc.
I'd like to add something to the (otherwise nice) answer of @BarocliniCplusplus. Although climatology used to be focused solely on the atmosphere, now it integrates other components of the climate system, such as the oceans, the vegetation, and even the crust. In that sense, one could say that "atmospheric science" focuses on one aspect of climatology (probably the main one, to be honest), but that "climatology" is not restricted to the atmosphere: you can be a climatologist and study ocean circulation.