Without context, it would be difficult to answer your question satisfactorily. A good dictionary will show common uses for both verbs.
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A health care professional told me the difference between quitting and stopping an addiction is that stopping simply means not doing it as in stopping smoking long enough to walk through a no smoking area. In contrast, quitting means not doing it is self imposed as in an ex smoker walks into a tobacco shop with a smoking friend and walks out still an ex smoker. If a smoker is on a sail boat an