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. 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!

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!