Basically they are the same... BUT there are some differences. Mistake can be a noun or a verb. I made a mistake. I mistook the woman with red hair for my wife! But I was mistaken! It was a complete stranger and I had to apologize for my mistake. Error is only a noun. I made an error. There is a verb... to err(to be mistaken or to do wrong/sin). There is no verb 'to error'. Usually the difference is in the collocations; just as fish goes with chips! 'I made an error in judgement' is a saying we have. You could say I made a mistake in my judgement... but the common expression is 'error in judgement'. If someone develops bad habits because they are influenced by other people... we say they were 'led into error'. We would NOT say they were 'led into mistake'. It does not sound right! If we do something wrong we usually say "Sorry, I did that by mistake!" We could say "Sorry I did that by error!" but it's not so common. So, there are places and situations where they are fully interchangeable with no alteration of meaning. There are places, often in fixed expressions, where we use one but not the other. And there are places where we'd probably prefer one not the other. The best thing is to keep reading and get familiar with those expressions... and then you won't be led into ERROR or make MISTAKES/ERRORS or be MISTAKEN about things. I'm sure other people will come up with some more ideas about this.

error is more serious

Not much difference. I would say mistakes are a something that are made by a person. You would make a mistake in your addition. There is an error in your addition. You would use error then if you do not want to attribute it to an action. You could still say 'there is a mistake in the addition' but the implication is stronger that a person made the mistake.