What are the 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.

For reference:


Meteorology time scales: Seconds-Weeks
Climatology time scales: Months-decades


In summary:


Atmospheric science is a generic term describing the study of the atmosphere.
Meteorology is the study of the atmosphere, generally in the short term. Occupationally, it involves interfacing with the public. Research meteorologists study a vast number of topics.
Climatology studies the evolution of the atmosphere over long timescales, including how it may be changing. Some of the things that meteorologists may neglect, climatologists must consider.