  We show that if $h\in \mathbb{Z}[x]$ is a polynomial of degree $k \geq 2$
such that $h(\mathbb{N})$ contains a multiple of $q$ for every $q\in
\mathbb{N}$, known as an $\textit{intersective polynomial}$, then any subset of
$\{1,2,\dots,N\}$ with no nonzero differences of the form $h(n)$ for
$n\in\mathbb{N}$ has density at most a constant depending on $h$ and $c$ times
$(\log N)^{-c\log\log\log\log N}$, for any $c<(\log((k^2+k)/2))^{-1}$. Bounds
of this type were previously known only for monomials and intersective
quadratics, and this is currently the best-known bound for the original
Furstenberg-S\'ark\"ozy Theorem, i.e. $h(n)=n^2$. The intersective condition is
necessary to force any density decay for polynomial difference-free sets, and
in that sense our result is the maximal extension of this particular
quantitative estimate. Further, we show that if $g,h\in \mathbb{Z}[x]$ are
intersective, then any set lacking nonzero differences of the form $g(m)+h(n)$
for $m,n\in \mathbb{N}$ has density at most $\exp(-c(\log N)^{\mu})$, where
$c=c(g,h)>0$, $\mu=\mu(\text{deg}(g),\text{deg}(h))>0$, and $\mu(2,2)=1/2$. We
also include a brief discussion of sums of three or more polynomials in the
final section.
