This paper investigates two distinct approaches to the dynamic modeling of aerial continuum manipulators (ACMs): the decoupled and the coupled formulations. Both open-loop and closed-loop behaviors of a representative ACM are analyzed. The primary objective is to determine the conditions under which the decoupled model attains accuracy comparable to the coupled model while offering reduced computational cost under identical numerical conditions. The system dynamics are first derived using the Euler--Lagrange method under the piecewise constant curvature (PCC) assumption, with explicit treatment of the near-zero curvature singularity. A decoupled model is then obtained by neglecting the coupling terms in the ACM dynamics, enabling systematic evaluation of open-loop responses under diverse actuation profiles and external wrenches. To extend the analysis to closed-loop performance, a novel dynamics-based proportional-derivative sliding mode image-based visual servoing (DPD-SM-IBVS) controller is developed for regulating image feature errors in the presence of a moving target. The controller is implemented with both coupled and decoupled models, allowing a direct comparison of their effectiveness. The open-loop simulations reveal pronounced discrepancies between the two modeling approaches, particularly under varying torque inputs and continuum arm parameters. Conversely, the closed-loop experiments demonstrate that the decoupled model achieves tracking accuracy on par with the coupled model (within subpixel error) while incurring lower computational cost.