In the present study we analyze the model utilized by Fischer et al from both classical and quantum dynamical viewpoints. We show that cavity-mediated suppression and enhancement of the isomerization dynamics at specific cavity frequencies can occur for different choices of the initial vibropolariton wavepacket. The classical dynamics of the initial phase space densities corresponding to the quantum initial wavepackets exhibit qualitatively similar behavior with varying cavity frequency. Quantum mechanically, the effects can be rationalized in terms of the specific class of polariton states that participate in the dynamics of the initial state. Classically, modulo tunneling contributions, the appearance of specific structures in the phase space and the associated chaosorder-chaos transition appear to correlate well with the cavity-mediated modulation of the isomerization dynamics. We establish that the correspondence between the classical and quantum results arises from the fact that the phase space nature of the dominantly participating polariton states is consistent with the appearance of the key phase space structures. In addition, for the model sys-tem of interest, our results indicate that cavity-mediated suppression of isomerization can occur in dark cavities. However, in this case the classical-quantum correspondence is compromised due to the dominance of quantum tunnelling.