Phase-contrast imaging is an emerging technology that may increase the signal-difference-to-noise ratio in medical imaging. One of the most promising phase-contrast techniques is Talbot interferometry, which, combined with energy-sensitive photon-counting detectors, enables spectral differential phase-contrast mammography. We have evaluated a realistic system based on this technique by cascaded-systems analysis and with a task-dependent ideal-observer detectability index as a figure-of-merit. Beam-propagation simulations were used for validation and illustration of the analytical framework. Differential phase contrast improved detectability compared to absorption contrast, in particular for fine tumor structures. This result was supported by images of human mastectomy samples that were acquired with a conventional detector. The optimal incident energy was higher in differential phase contrast than in absorption contrast when disregarding the setup design energy. Further, optimal weighting of the transmitted spectrum was found to have a weaker energy dependence than for absorption contrast. Taking the design energy into account yielded a superimposed maximum on both detectability as a function of incident energy, and on optimal weighting. Spectral material decomposition was not facilitated by phase contrast, but phase information may be used instead of spectral information.