We present constraints on neutrino masses, the primordial fluctuation spectrum from inflation, and other parameters of the ACDM model, using the one-dimensional Ly alpha-forest power spectrum measured by [1] from the Baryon Oscillation Spectroscopic Survey (BOSS) of the Sloan Digital Sky Survey (SDSS-III), complemented by Planck 2015 cosmic microwave background (CMB) data and other cosmological probes. This paper improves on the previous analysis by [2] by using a more powerful set of calibrating hydrodynamical simulations that reduces uncertainties associated with resolution and box size, by adopting a more flexible set of nuisance parameters for describing the evolution of the intergalactic medium, by including additional freedom to account for systematic uncertainties, and by using Planck 2015 constraints in place of Planck 2013. Fitting Ly alpha data alone leads to cosmological parameters in excellent agreement with the values derived independently from CMB data, except for a weak tension on the scalar index n(s). Combining BOSS Ly alpha with Planck CMB constrains the sum of neutrino masses to Sigma m(nu) < 0.12 eV (95% C.L.) including all identified systematic uncertainties, tighter than our previous limit (0.15 eV) and more robust. Adding Ly alpha data to CMB data reduces the uncertainties on the optical depth to reionization tau, through the correlation of tau with sigma(8). Similarly, correlations between cosmological parameters help in constraining the tensor-toscalar ratio of primordial fluctuations r. The tension on n(s) can be accommodated by allowing for a running dn(s)/d In k. Allowing running as a free parameter in the fits does not change the limit on Sigma m(nu). We discuss possible interpretations of these results in the context of slow-roll inflation.