Consider the map $\varphi:S\to R$ given by
$$\varphi(x) = \frac{4x}{4-x},$$
and similarly for $y$, $z$, and $w$. One needs to check that this is well-defined;
indeed, $\varphi$ maps $xyz+xyw+xzw+yzw+xyzw$ to
$$\frac{4^4(xyz+xyw+xzw+yzw)}{(4-x)(4-y)(4-z)(4-w)}.$$
Since $\varphi$ obviously induces an isomorphism on the associated graded of the filtration by powers of the maximal ideal, it is itself an isomorphism.
More explicitly, the inverse homomorphism $\psi:R\to S$ is given by
$$\psi(x) = \frac{4x}{4+x},$$
and similarly for $y$, $z$, and $w$.
Note also that this generalizes to any number of variables.
Thanks David and Vladimir for your answers; that was really helpful for getting me going!