I was reading the two posts in the references to understand the distinction between polynomials and polynomial functions, yet I felt that I couldn't find the conclusion... So let me ask here.
Is it correct to rephrase the above comparison as polynomial expressions and polynomial functions as a polynomial is an expression of relations of variables and a polynomial function actually allows us to evaluate it on some value in a field of domain.
For instance, from @GitGud, suppose two polynomial functions $p(x) = x^2 + 1$ and $q(x) = x+1$ on $\mathbb{F} = \{ 0, 1 \}$, then $p(x) = q(x)$ as $p(x=0) = q(x = 0)$ or $p(x=1) = q(x = 1)$. But as expressions, $p \neq q$.
