It is a known fact that if you repeatedly take the forward finite difference of a series of $n$th powers $n$ times, you obtain a sequence whose only elements are $n!$ ($n$ factorial). For example, the series of cubes:
\begin{align*} &0, 1, 8, 27, 64, 125, 216, 343...\\ &1, 7, 19, 37, 61, 91, 127...\\ &6, 12, 18, 24, 30, 36...\\ &6, 6, 6, 6, 6... \end{align*}
I recently rediscovered this for myself, and my searches to see what was known about it led me to this stack exchange question (where there are many more examples and demonstrations of this fact). That question has two answers, both of which give entirely satisfying rigorous proofs of why this pattern exists.
However, I still can't scratch that intuitive itch, as I don't understand, from an arithmetic standpoint, what the factorials have to do with the perfect powers. After all, their definitions are simple (on the positive integers): we (intuitively) define $x^n$ as $x$ multiplied by itself $n$ times, and we (intuitively) define $n!$ as all the positive integers below and including $n$ multiplied together. Besides both being related to repeated multiplication, I don't see any similarities that would give an intrinsic explanation.
In this 3blue1brown video, a non-mathematician asks her statistician friend, "What do circles have to do with population?" (upon seeing the appearance of $\pi$ in a population formula). I feel that my question is in a similar vein; I understand the rigorous proof for why the factorials underlie the perfect powers, but not the intuition behind it. I'm looking for answer that explains this as if I were a non-mathematician -- there's no need for a rigorous answer, just intuitive.
So, after all that:
Why, intrinsically, do the factorials underlie the perfect powers?
