a) Electronic mail: jfoley19@charlotte.edu Although these approaches are equivalent in their exact limits, it is practically impossible to reach these exact limits for the vast majority of physically relevant systems. Outside of their exact limits, the variety of approximations inherent in electronic structure calculations makes it quite difficult to assess these two approaches on equal footing. In this work, we attempt to provide a fair assessment of these two approaches based upon simple analysis of their underlying formalisms, and through comparable numerical results approaching exact limits. To this end, we implement variational approaches to pQED and sc-QED using full configuration interaction (FCI) to parameterize the former and a self-consistent QED-FCI scheme for the latter. Analysis of the formalism of both approaches shows that these approaches are indeed equivalent in the limit of a complete orbital and many-electron basis, but have a key disparity in a quadratic light-matter coupling term, known as the dipole self energy, outside of this limit. While scQED utilizes the exact form of the dipole self energy operator in a given orbital basis, the projected dipole self energy that arises in pQED is inexact except for in a complete orbital and many-electron basis. We present numerical results highlighting this disparity and its resolution in a particularly simple molecular system of helium hydride cation, where it is possible to approach these two complete basis limits simultaneously. In this same helium hydride system, we examine and compare the practical issue of computational cost required to converge each approach towards the complete orbital and many-electron bases limit. Finally, we assess the aspect of photonic convergence for polar and charged species, finding comparable behavior between parameterized and self-consistent approaches.