The effects of surface irregularities and imperfections on the thermal resistance at a solid-liquid interface have been investigated using molecular dynamics. The molecular model comprises liquid argon confined between silver walls. The surface roughness was designed using fractal theory, introducing stochastic patterns of multiple scales that resemble realistic surface geometries. In agreement with most previous studies, we find that increasing the strength of the solid-liquid interactions monotonically reduces the thermal resistance across smooth interfaces. Yet, the behavior of the thermal resistance across rough surfaces is more complex. Following the initially anticipated decrease, the thermal resistance starts to increase once the strength of solid-liquid interaction increases past a threshold. We attribute the above behavior to two competing phenomena, namely, the area of the solid-liquid interface and the introduction of vibrational anharmonicities and localization of phonons resulting from the surface roughness. Finally, we demonstrate that, for the same fractal dimension and depth of surface roughness, different surfaces practically have the same thermal resistance, solid-liquid radial distribution function, and liquid density profiles. We conclude that the above fractal parameters are useful in deriving reduced models for properties related to the surface geometry.