playBunny: I checked both words, error and mistake, with my Oxford Advanced Learner's Dictionary amd this is what it says:

MISTAKE: 1.An action or opinion that is foolish or wrong; and error in judgement.
                     2.A word, figure, sum, etc.,that is not correct.


Error: 1.A thing done wrongly, a mistake.
           2.The state of being wrong in belief or behaviour.


And this is what I just now found on the next page.  I hadn't noticed it before:

Note: Compare mistake, error, blunder, fault and defect.
         
They all refer to something that has not been done correctly or properly.  Mistake is the most      general and used in most situations: The letter had quite a few mistakes in it. Error is used when talking about calculations and in technical or formal contexts: I think there are a few errors in your calculations.  A blunder is a stupid or careless and quite serious mistake made because of bad judgement: A hospital blunder led to 500 cancer patients getting the wrong radiation treatment.  Fault emphasizes a person's responsibility for a mistake: Tom broke the window but it was my fault for letting him play football in the house.  A fault can also be an imperfection in a person or thing: There was a design fault in the train doors.  A defect is a serious imperfection:  The causes of many birth defects have not yet been discovered.

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I cannot fully understand the meaning of this sentence:

"Accidents are caused by errors. (Though you could mistake something in
error, lol, eg the pilot error could be that he mistakenly pressed the
ejector seat button instead of the seat adjustment button.)"

What do you mean 'you could mistake something in error?'

If I understand it correctly, you may mean that here 'error' has a stronger meaning than 'mistake,' but I'm not sure.

Thanks for the etymology of the two words.  One of the main problems that I have with English is choice of vocabulary.  Here's an example:

I see a very beautiful scene.  I want to show how much I'm impressed with it.  There are many words that I can use:

Wonderful, magnificent, fabulous, incredible, marvelous, fascinating, unbelieveable, terrific,etc..

Now suppose I use 'wonderful.'  We move on and we see a more beautiful scene.  Now I have a problem to deal with.  I no longer know which word to choose because I don't know which one shows that I am more impressed with this scene than the previous one!

Or, when there are some words that I think have the same meaning, I am not sure which one to choose, because they may not really have the exact same meaning:

Probably, perhaps, possibly, maybe, may, might, likely.

Like I don't know if I see the sky's cloudy, which one of the following sentences I can use:

• Probably it'll rain
• Perhaps it'll rain.
• It'll possibly rain.
• Maybe it'll rain.
• It may rain.
• It might rain.
• It's likely to rain.

Well, I guess I will never be able to solve these problems unless I live among native speakers.  When I see what they use in what situation, I can learn that too!