It depends on the property you are interested in. For instance, in magnetism, magnetic nanowires exhibit shape anisotropy, which can help to stabilize the magnetization along the wire axis. In plasmonics, noble metal nanowires exhibit different plasmonic resonances depending if the light is polarized along the axis or perpendicular to it, and the localization of the electromagnetic field can be enhanced in those resonances.
i think it depends on its specific application, one cannot discribe the advantages of nanorod without involving its application. For example, researchers try to synthesize metal nanorod with a high exposure of specific crystal facet, which can improve catalytic efficiency.