Exception and error are two different words which people understand to be different in their context. An error can be said to be a flawed approach, behavior, or answer to a particular situation base on the set down rules, laws, or norms involved in the scenario. An exception is like a mistake that can still be amended. It is a situation that is almost tending to error, but it is not really wrong in its behavior or approach.
If a mistake is made or someone erred without any pressurized reason, that is an error, but if there is an eminent maybe negative occurrence that has pushed him or her into making a mistake, then it can be seen as an exception. For example, a man driving beyond the speed limit on the road because of carelessness or negligence is making an error. But if a man whose wife is in great pain, while trying to rush her to the hospital over speed, that can be seen as an exception.