Considering the requirements of the actual production scheduling process, the utilization of the genetic programming hyper-heuristic (GPHH) approach to automatically design dispatching rules (DRs) has recently emerged as a popular optimization approach. However, the decision objects and decision environments for routing and sequencing decisions are different in the dynamic flexible job shop scheduling problem (DFJSSP), leading to different required feature information. Traditional algorithms that allow these two types of scheduling decisions to share one common feature set are not conducive to the further optimization of the evolved DRs, but instead introduce redundant and unnecessary search attempts for algorithm optimization. To address this, some related studies have focused on customizing the feature sets for both routing and sequencing decisions through feature selection when solving single-objective problems. While being effective in reducing the search space, the selected feature sets also diminish the diversity of the obtained DRs, ultimately impacting the optimization performance. Consequently, this paper proposes an improved GPHH with dual feature weight sets for the multi-objective energy-efficient DFJSSP, which includes two novel feature weight measures and one novel hybrid population adjustment strategy. Instead of selecting suitable features, the proposed algorithm assigns appropriate weights to the features based on their multi-objective contribution, which could provide directional guidance to the GPHH while ensuring the search space. Experimental results demonstrate that, compared to existing studies, the proposed algorithm can significantly enhance the optimization performance and interpretability of energy-efficient DRs.