There are various definitions for both words. The difference between the two depends on the context in which both are used. They can also have the same meaning.
I made a "mistake" and broke my toe.
I was "wrong" when I slapped him for no reason.
I made a mistake and gave the money to the man in brown.
Do not mistake him for his twin.
I was wrong to take my problems out on you.
That problem is wrong.
You went the wrong way.
Marrying her was a mistake.
I said I had $5.00, but I was wrong.
"Two wrongs do not make a right."
Do not wrong your brother.