This study explores a nonlocal reaction-diffusion-advection system that models interactions between two competing phytoplankton species in a water column, incorporating crowding effects. We introduce a special cone 𝕂 based on the cumulative distributions of population densities and establish a comparison principle under the order induced by𝕂. This results in strong monotonicity within the semiflow generated by the system. We then analyze the dynamics of the system in terms of the advection rates of the two phytoplankton species using monotone dynamical system theory. We identify critical curves that categorize competition outcomes into competitive exclusion, coexistence, and/or bistability. The position and shape of these critical curves can vary significantly depending on key parameters, such as death rates. Furthermore, we derive global results for specific scenarios using a perturbation approach. These findings highlight the crucial role of advection rates and death rates in shaping dynamics within two-species phytoplankton communities.