There are appropriate ways to use these words ( mistakes & errors ), and this will often depend on the context.
‘Error’ and ‘Mistake’ are two of these words. Both of the words mean: “A wrong action attributable to bad judgment, or ignorance, or inattention”. Many use these words interchangeably, which can be right for certain situations, but some would deem a particular word as more appropriate than the other.
As what has been said earlier, the context will dictate the proper usage. ‘Error’ and ‘mistake’ fall into the same category. Many say that ‘error’ is more severe. It is due to miscalculation and wrong judgment, that ‘mistake’, on the other hand, is less in gravity, as people normally make mistakes. However, there are also many people who will argue with this dissection.
It is highly acceptable to use ‘error’ in formal or technical contexts. In scientific or highly technical terms, the word ‘error’ is more suitable. In the world of computing and programming, ‘error’ is the more fitting term to indicate a mistake, or fault, particularly in coding and processes. ‘System Error’ sounds better than ‘System Mistake’, doesn’t it?