What is exactly the difference between a definition and an axiom? I am wondering what the difference between a definition and an axiom.
Isn't an axiom something what we define to be true?
For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the successor of any natural number. Why can't we define 0 to be the natural number such that it isn't the successor of any natural number?
In general, why is an axiom not a definition? 
Are there cases where we can formulate definitions as axioms or vice versa?

I'm not sure whether this answer would help but if you see the axioms and definitions listed in this page you might get to know what's the difference. Axioms acts as fundamentals while definitions are statements that include axioms to say about something.
