The Goemans-Williamson algorithm in the $SOS$ framework
Looking at the Goemans–Williamson algorithm in the SOS framework yields no technical advantages: it is exactly the same algorithm and the same ideas are used in the analysis.
The only advantages in doing so are:


Arguably the algorithm seems less "magical" in that viewpoint, though of course that's a matter of taste.
It's a good basic case to get intuition for the SOS framework, and so understanding this algorithm will help in understanding more complicated uses of it.


As Sasho Nikolov points out, a degree 2 pseudo-distribution is simply a positive semidefinite (PSD) matrix, and so finding such a matrix satisfying any particular linear constraints can be done via semidefinite programming. In particular, for any pseudo-distribution $\{ u_i \}_i$ of degree at least 2, the matrix $X_{i,j} = \tilde{\mathbb{E}} u_i u_j$ is PSD. 

Given an $n\times n$ PSD matrix $X$, one can find a Gaussian distribution $y\in\mathbb{R}^n$ such that $\mathbb{E} y_iy_j = X_{i,j}$ for all $i,j$ and hence the distribution $\{ y_i \}_i$ matches the second moments of the pseudo-distribution represented by $X$. (One can obtain $y$ by sampling a standard Gaussian vector $z$ and letting $y=X^{1/2}z$.)

The rounding algorithm is to obtain a distribution $w$ over $\{\pm 1\}^n$ by letting $w_i = \textrm{sign}(y_i)$. This is exactly equivalent to the standard description of "hyperplane rounding" whereas one takes vectors $v_1,\ldots,v_n$ such that $X_{i,j} = \langle v_i,v_j \rangle$ and lets $w_i = \textrm{sign}(\langle v_i , z\rangle)$ where $z$ is a random Gaussian vector. 

Now we want to compare the value of the pseudo-distribution, which (up to scaling) is the same as the sum of $-X_{i,j}$ over all edges $(i,j)$, with the expected value of our rounded solution $w$, which will be the sum of $- w_iw_j$. So now we want to calculate the probability that the edge will be cut by $w$ as a function of $X_{i,j}$. It all boils down to figuring out the following calculation: if $w_i$ and $w_j$ are two Gaussian random variables with $\mathbb{E} w_i^2 = \mathbb{E} w_j^2 = 1$ and  $\mathbb{E} w_i w_j = \rho$, what would be the probability that their signs agree? I personally think that the most interesting regime for Max-Cut is when the objective value is close to $1$. One nice thing in that regime is that you can get some qualitative insight by showing that if $\rho = -1 +\epsilon$ then their signs agree with probability at most $O(\sqrt{\epsilon})$. However, if you are interested in figuring out the approximation ratio (or more generally figure out the precise trigonometric function behind this $O(\cdot)$ term) then Goemans–Williamson's analysis exactly resolves this calculation.