I think "error" is the more general one, and "mistake" is when it's done by someone. You can make a mistake, but your computer can't - those are errors. (I've also read that "bug" and "glitch" have a similar difference. I think it's that "glitch" is the symptom, and "bug" is the cause - but the first computer "bug" was a literal bug mucking up the works, so it would really be a glitch?) That's the way I use it, at least. I think then, when "mistake" is applicable, it's a different connotation if you just say "error" - I think "error" would then focus on the consequences, as they differ from how it should be, while "mistake" makes it more personal. I think in the specific case of a spelling error, unless I'm familiar with the process of how the text came to be, I can't really know if it was the author or the "printer" that caused the error. Actually, I first typed "made the error" for the previous sentence, but that looked wrong, so I think it's really that "error" denotes the effect, while "mistake" is the glitch in the human that is erroneous.
I also agree with oddtail's example. I'm starting to get the impression that "error" is also more abstract, while "mistake" is more tangible. Back with the spelling error - if I caused the spelling error, I use the word "mistake" in my mind, but if I see that something else caused it, like a glitch in the computer, then I use "error". Similarly, I don't know about how the error happened in the case of seeing someone else's writing having poor spelling, so I use "error" for that.
IICIH? I think the avionics one is because "err" originally meant "wander" so the "error" would be how much the flight has strayed from the precise course, which I understand is expected and natural in flight.