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Pros: With trigonometric basis functions your problem size is $N \text{log}(N)$ instead of $N^2$. Stabilization techniques are easy to implement and cheap: Filtering in the modal space. Zero padding in the modal space. No aliasing due to the Galerkin ansatz. Energy/Entropy stable disctretizations, e.g. via a skew symmetric implementation, are quite easy. ...
To expand on Wolfgang Bangerth's answer, I think P0 DG schemes reduce to two-point cell-centered finite volume schemes. I don't know if DG convergence analysis always includes $p = 0$, but the resulting finite volume schemes can be shown to converge under appropriate "mesh orthogonality" conditions. https://math.unice.fr/~minjeaud/Donnees/...