We report here the effects of chain stiffness and surface attachment on the effective interactions between polyelectrolyte-grafted colloidal particles in monovalent salt obtained using Monte Carlo simulations. Our approach involves computation of the distance-dependent potential of mean force between two polyelectrolyte-grafted colloidal particles treated at a coarse-grained resolution. Two chain stiffnesses, flexible and stiff, and two surface attachment modes, free and constrained, at low grafting densities are examined. PMF calculations across a range of surface and polyelectrolyte charge allows us to map out the strength and extent of the attractive and repulsive regime in the two-dimensional charge space. We observe striking differences in the effects of chain stiffness between the two modes of attachment. When the chains are freely attached, the stiff-chains colloids exhibit a marginal reduction in the attraction compared to their flexible-chain counterparts. In contrast, when the chains are attached in a constrained manner, the colloids with stiff chains exhibit a significantly stronger attraction and a broader attractive regime compared to flexible-chain colloids. These differences in the effects of stiffness between the two attachment modes are explained in terms of differences in the energetic and entropic forces balancing adsorption of chains at their own surface versus chain extension to mediate bridging interactions across two particles. Our results thus underscore the importance of surface attachment of chains and its proper accounting in computational and experimental studies and suggests the mode of chain attachment as an additional control parameter for modulating intercolloid interactions for applications such as stabilization of colloidal systems and bottom-up self-assembly of nanostructures.