What are the benefits of using spectral k-means over simple k-means?

Question

I have understood why k-means can get stuck in local minima.

Now, I am curious to know how the spectral k-means helps to avoid this local minima problem.

According to this paper A tutorial on Spectral, The spectral algorithm goes in the following way

1. Project data into $R^n$ matrix

2. Define an Affinity matrix A , using a Gaussian Kernel K or an
   Adjacency matrix

3. Construct the Graph Laplacian from A (i.e. decide on a normalization)

4. Solve the Eigenvalue problem

5. Select k eigenvectors corresponding to the k lowest (or highest) eigenvalues to define a k-dimensional subspace

6. Form clusters in this subspace using k-means

In step 6, it is using k-means. How is it overcoming the local minima problem of k-means? Moreover, what are the benefits of spectral over k-means?

If someone gives detailed insight upon this, it will be very helpful for me.