A function f(x) is polynomially larger than g(x) if it is greater only by some polynomial factor e.g. by n2 , n3 but the exponent should be some constant not a function of n since then it will be even larger than factorial.
A function is asymptotically larger if it follows big -Oh notation .This is necessary and sufficient condition and here f(x) can be larger than g(x) by any factor , not necessarily polynomial.Let us understand what I have said through examples.
If f(x) = n5 and g(x) = n3logn , so removing common of n3 both sides , we have :
n2 which is polynomially larger than logn
So here f(x) is polynomially larger than g(x)
But if we take f(x) = nn and g(x) = n2
So now f(x) is not polynomially larger but we can say exponentially larger and to be generic asymptotically larger.
If we consider the other side , f(x) = nlogn and g(x) = n
Here also it is not so because the factor is logn which is less than polynomial but we can say asymptotically larger.
I hope you have understood the difference between the two.