Monometallic platinum and nickel nanoparticles, and platinum-nickel nanoalloys are examined in the range 13-976 atoms from density functional theory calculations. A large set of competitive symmetries and morphologies are considered including the usual Mackay icosahedral, Marks-decahedral and truncated octahedral forms. A comparative analysis of relative stability order is addressed on the basis of four stability descriptors all predicted at the ab initio level from spin-polarized calculations including van der Waals interactions. For platinum nanoparticles, they unanimously conclude on the preference of truncated octahedral morphology in the range 147-201 atoms. For nickel and platinum-nickel nanoclusters, three descriptors (cohesion energy, nanoparticle surface energy and vibrational band center) also support such octahedral symmetry (with a skin-heart chemical ordering for nanoalloys), whereas the excess energy rather favors the icosahedral morphology (with multishell and core-shell arrangement). Such discrepancies feed the debate related to the impact of the normalization on the predictive power of these descriptors and recall the high importance of validating theoretical models from a quantitative standpoint. This work invites the experimentalists to synthesize, characterize and measure surface energetics of PtNi nanolloys in highly controlled operating conditions.