We have carried out atomistic molecular dynamics simulations of a melt of 1,4-poly(butadiene) from temperatures well above the experimentally observed merging of the primary alpha process and secondary beta process down to temperatures approaching the experimentally observed bifurcation temperature. The relaxation strength and maximum loss frequency and its temperature dependence for the combined alpha-beta dielectric relaxation process from simulations were in good agreement with experiment. The maximum loss frequency, melt viscosity, chain normal-mode relaxation times and torsional autocorrelation times were found to exhibit nearly identical non-Arrhenius temperature dependencies well represented by a Vogel-Fulcher fit with parameters in good agreement with experimental values obtained from dielectric and viscosity measurements. The dielectric susceptibility showed increasing intensity at high frequency for the lower temperatures investigated, indicative of a breakdown in time-temperature superposition due to an emerging beta process. Comparison of time scales for the chain normal-mode dynamics and dielectric relaxation revealed that the latter is associated with motions on the segmental length scale. The correspondence of time scales and temperature dependence for the dielectric relaxation and the torsional autocorrelation function further confirmed the localized nature of the dielectric relaxation and indicated that the combined alpha-beta dielectric process is fundamentally tied to microscopic conformational dynamics of individual dihedrals. However, the mean conformational transition rates were found to exhibit Arrhenius temperature dependence, leading to a divergence of time scales between the torsional, dielectric, chain and mechanical relaxation processes and the rates of conformational transitions with decreasing temperature. This divergence was associated with the increasingly heterogeneous character of conformational dynamics in the melt with decreasing temperature. Hence, the time scale of the principal (alpha) relaxation in the melt is fundamentally correlated with the time scale for homogenization of conformational dynamics, and not to the time scale of the conformational transitions themselves.