They mean the same thing pretty much, but can be used differently depending on the context.
Error: "a deviation from accuracy or correctness; a mistake, as in action or speech: His speech contained several factual errors. "
"to err (to make mistakes) is human," "the dark spot on the x-ray was nothing more than an error," "I now see the error of my ways."
Mistake: "1.
an error in action, calculation, opinion, or judgment caused by poor reasoning, carelessness, insufficient knowledge, etc.
2.
a misunderstanding or misconception. "
Y,ou can also mistake someone, and, alternatively, mistook someone.
"The typo on the front page was my mistake," "I made the mistake of not bringing an umbrella," "I think you've mistaken me for someone who gives a damn." mis-taken (miss: wrongly, taken: assumed. Best way I can explain that one.)
"I mistook you for someone else."
Sorry for all the examples. I just love doing them. So you see, mistake and error are very similar in meaning. (I added in mistook just because it's one of my favorite words lol) As always, if anyone should see that I've made an error, please correct me. :) Hope this helps at least a little.