Organic additives are known to alter the mineral-water interface in various ways. On the one hand, organic molecules can self assemble into ordered structures wetting the surface. On the other hand, their presence can affect the interfacial morphology, referred to as surface restructuring. Here, we investigate the impact, of a class of calcium-complexing azo dyes on the dissolution of calcite (10.4) using high-resolution atomic force microscopy operated in aqueous solution, with a focus on the two constitutional isomers Eriochrome Black T and Eriochrome Black A. A very pronounced surface restructuring is observed in the presence of the dye solution, irrespective of the specific dye used and independent of the pH. This surface restructuring is obtained by the stabilization of both the nonpolar acute and the polar [010] step edges, resulting in a greatly altered, characteristic interface morphology. In sharp contrast to the prevalence of the surface restructuring, an ordered molecular structure on the crytal terraces is observed only under very specific conditions. This formation of an ordered stripe-like molecular structure is obtained from Eriochrome Black A only and limited to a very narrow pH window at a pH value of around 3.6. Our results indicate that such molecular self-assembly requires a rather precise adjustment of the molecular properties including control of the conformation and deprotonation state. This is in sharp contrast to the additive-induced surface restructuring, which appears to be far more robust against both pH changes and variations in the molecular conformation.