We theoretically and systematically investigate Anderson localization of two bosons with nearest-neighbor interaction in one dimension under short- and long-time scales, two types of disorders, and three types of initial states, which can be directly observed in linear disordered photonic lattices via two experimentally measurable physical quantities, participation ratio and spatial correlation. We find that the behavior of localization characterized by the participation ratio depends on the strength of interaction and the type of disorder and initial condition. Two-boson spatial correlation reveals more novel and unique features. In the ordered case, two types of two-boson bindings and bosonic "fermionization" are shown, which are intimately attributed to the band structure of the system. In the disordered case, the impact of interaction on the two-boson Anderson localization is reexamined and the joint effect of disorder and interaction is addressed. We further demonstrate that the independence of the participation ratio or spatial correlation on the sign of interaction can be eliminated by employing an initial state that breaks one of two specific symmetries. Finally, we elucidate the relevant details of the experimental implementation in a two-dimensional linear photonic lattice.