Mistake (noun) = something that was done or said the wrong way.
e.g., I made a mistake when I told her that her new dress made her look fat. :)
Fault (noun) = the thing or person that is responsible for the problem or mistake.
e.g., It's my fault that she got angry at me. I should have thought before I said she looked fat. I am to blame for making a bad mistake. :)
I DID OR SAID SOMETHING THAT WAS INCORRECT / IMPROPER / HURTFUL / DISRESPECTFUL / INCONSIDERATE.
Those are all mistakes.
I AM THE ONE WHO IS RESPONSBLE FOR SAYING OR DOING SOMETHING THAT WAS INCORRECT / IMPROPER / HURTFUL / DISRESPECTFUL / INCONSIDERATE.....
It is my fault that I made those mistakes. I can't blame anyone else. I was wrong, so those mistakes (wrong thoughts or actions) are my fault (because of me and no one else).
If there are any mistakes in my explanation above, they could be my fault, but I wish I could blame the mistakes/errors on my computer or maybe even Microsoft :)