What's the difference?
This one is not so clear as both words have two similar meanings.
So let’s begin with the first set of definitions of mistake and error.
Mistake: To do something – usually by accident - wrong or incorrect. She made too many mistakes in the test. We can learn by our mistakes. This is the third time you’ve made the same mistake.
(You should know this, but wrong and incorrect are adjectives, not nouns. We're dealing with nouns here.)
An error is a mistake that you do not realise you are making, but could cause serious problems. It is often used in more formal situations and is perhaps more euphemistic than mistake: A computer error resulted in my receiving an electricity bill for ten thousand pounds! Human error resulted in the plane crash.
My handy copy of the Longmans Language Activator has some other synonyms based on this definition of error (a mistake that you do not realise you are making, but could cause serious problems):
slip (a small unimportant mistake)
mix-up (a careless mistake that causes confusion)
slip-up (a mistake that spoils a plan, process, or something you want to achieve)
oversight (a mistake because you didn’t notice or forgot to do something)
aberration (a mistake in something that isn’t normal and is unusual).
The second set of definitions of mistake and error
A mistake is to do something wrong which can be made in speaking and writing: If you make a mistake, put an ‘X’ on the number you chose and put a circle around another choice. One common mistake by students is not using ‘a’, ‘an’ or ‘the’.
Error is also a mistake that you don’t realise, or that you think is correct but is not. It’s very easy for a translator to make errors in written texts. Students should be allowed to correct their own errors.
For synonyms here, we have:
misprint (a word spelt wrongly on a printed page)
spelling mistake (spelling a word wrongly)
slip of the tongue (saying a different word which is wrong and could be a mistake that you really don’t want to make).
You will find that mistake has several meanings, but the ones listed above are the ones that are most commonly confused with error.