Method of Moments is the name given to the Boundary Element Method (BEM) in the Electromagnetism community. Since you are using Green functions for BEM you get fully populated and asymmetric matrices.
Some advantages that are commonly mentioned for BEM are:
It reduces the dimensionality of the mesh. This might look to lead to smaller matrices but, in general, they are more expensive compared to sparse matrices for FEM. This might be an advantage when dealing with remeshing such as fracture mechanics and shape optimization.
You can deal naturally with unbounded domains, such as exterior acoustics, antenna radiation, or seismic wave propagation.
You can add Fast Multipole Method to improve BEM performance. And, I would say that is the way to go instead of just using plain BEM nowadays.
Liu, Y. (2009). Fast multipole boundary element method: theory and applications in engineering. Cambridge university press.