The elastic and piezoelectric properties of zincblende and wurtzite crystalline InAs/InP nanowire heterostructures have been studied using electro-elastically coupled continuum elasticity theory. A comprehensive comparison of strains, piezoelectric potentials and piezoelectric fields in the two crystal types of nanowire heterostructures is presented. For each crystal type, three different forms of heterostructurescore-shell, axial superlattice, and quantum dot nanowire heterostructuresare considered. In the studied nanowire heterostructures, the principal strains are found to be insensitive to the change in the crystal structure. However, the shear strains in the zincblende and wurtzite nanowire heterostructures can be very different. All the studied nanowire heterostructures are found to exhibit a piezoelectric field along the nanowire axis. The piezoelectric field is in general much stronger in a wurtzite nanowire heterostructure than in its corresponding zincblende heterostructure. Our results are expected to be particularly important for analyzing and understanding the properties of epitaxially grown nanowire heterostructures and for applications in nanowire electronics, optoelectronics, and biochemical sensing.