If I am talking about doing some normal action, start and begin are synonyms, there is no difference. "I began/started doing the housework." "I began/started my studies two years ago." However, I would start a car, motor or any machine. I would not begin them. Also, I would tend to use 'start' if something went from stop to going very fast. "The runners started at the gun." - 'Begin' would not be used here.

They have the same meaning, no difference in usage. 'Start' is slighty more common then 'begin'.