Under the simplifying assumption of perfect adhesion, motivated by the wide adoption of highly resistant glues, delamination of FRP-reinforced masonry pillars turns out to be governed exclusively by the non-linear behavior of the quasi-brittle, heterogeneous support. However, at the structural level, the macroscopic delamination response can be described effectively by concentrating all the sources of dissipation and non-linearity at the masonry-FRP interface, and assuming the support to behave as a linear-elastic body. In this paper, a detailed and critical comparison between two different fully three-dimensional finite element models is developed: (i) a model in which only masonry (i.e. brick and mortar independently) is damageable whilst the FRP reinforcement adheres perfectly to the support (namely, exclusively bulk damage is accounted for), and an alternative model (ii) in which a cohesive, zero thickness interface between the FRP and the support is considered (i.e. interface damage), whilst masonry behaves as a heterogeneous linear elastic material. The overall response during delamination and local stress distributions at the interface are critically investigated, varying the FRP reinforcement width.