The math notation doesn't need parens. The computer notation needs parens. This alone is simple obvious proof that they are different notations, at least in my understanding of what a notation is.
-3^2 in the math notation doesn't need parens only because of operator precedence.
The fact that the computer ^ requires parens in more cases like -3^(2+2) is irrelevant for this and doesn't allow you justifying different precedence rules (and your downgrading from "completely different" to "different" isn't a proof, just "tautology". Hey, they also look different, so they are different!)
