An error and a mistake, although share similar denotation, truly do vary in connotative meaning. An error is something wrong that is done when not knowing any different, versus a "mistake" implies something wrong that is done when knowing better. I'm not strong in math, and I typically would make errors on exams since I didn't have control of higher level math rules. However, I make careless mistakes when balancing my check book because I really despise doing it. There is a definite difference in connotation.
As far as error analysis, this is the formal study or examination of the types and causes of errors. In what ways do non-native speakers make errors in language use: pronunciation, writing, listening, and choice of vocabulary? This is, of course, a much different question when it comes to discussing the common mistakes made by native speakers. Or a bigger question may be, in what ways do native speakers make errors in language use? Are these "errors" or are they truly "mistakes"? When a student says or writes, "I ain't got none", is it a statement made due to a lack of knowledge or is it knowlingly a statement which breaks the rules due to laziness and habit? I think we know the answer.
Error analysis, in linguistic terms, informs us that the difference in an error and a mistake lies in the fact that, with language acquisition, errors are systematic whilst mistakes are not. What this means is that those who learn to acquire a language, specifically as a second language, will make language errors as a rule, whether they be errors in pronunciation, spelling, grammar, punctuation and so forth. In other words, these errors will systematically occur since they are part of the process of acquiring the language.
Obviously, many, most, or even all of these errors will cease to occur the more competent and confident the student becomes in the language he or she is in the process of acquiring. Furthermore, it should be easy to establish why the student makes these errors if one compares the student's home (first) language to the one being acquired. It should become evident that the influence of the pronunciation and other conventions of the home language is what causes the error/s.
A mistake, on the other hand, is not caused by such influences. Mistakes in language are not, in terms of the above definition, caused because of the influence of an already acquired language but are the result of oversight, carelessness, poor understanding, and so forth. A mistake is not caused by the conventions of another language. One can, for example, understand the error made by a second language speaker if he should use a double negative in a sentence if this is a convention in his home language.
An Afrikaans (a South African language) speaker for example, could say in English 'I did not know not' since the double negative is used in his language. In addition, he may say, 'You is my brother' since there is no distinction between singular and plural verbs in Afrikaans. Because of this systematic recurrence, the wrong grammar use is an error and not a mistake.
The implication, in linguistic terms therefore, is that First language speakers make mistakes in their speaking, writing and reading since their poor language skills cannot be deemed systematic and may be the result of other factors such as poor level of education or the unwillingness to learn.
However, "although error analysis is still used to investigate specific questions in Second Language Acquisition, the quest for an overarching theory of learner errors has largely been abandoned. In the mid-1970s, Corder and others moved on to a more wide-ranging approach to learner language, known as interlanguage."