Taking into account detailed chemical kinetics and therefore allowing for a detailed representation of the flame's microstructure at reduced computational cost make flamelet-based tabulation approaches such as the flamelet-generated manifold (FGM) a commonly used method for turbulent combustion simulations. However, there has been little focus on analyzing such models for fuel blends, including hydrogen. One reason for that is the challenging inclusion of differential diffusion effects into FGM, which may become crucial for highly diffusive fuels such as hydrogen. This paper presents an extension of the FGM approach that takes into account differential diffusion to assess the importance of differential diffusion for methane hydrogen blends. To this end, an extended model containing five controlling variables can be derived. However, the high correlation of certain controlling variables and the number of control variables could be reduced to three controlling variables in this study. These models are coupled to the artificially thickened flame (ATF) approach to facilitate large-eddy simulations (LESs). To ensure the consistency of the coupling between FGM and ATF when differential diffusion is considered, the model is thoroughly verified and validated using freely propagating and stratified laminar one-dimensional flames. Finally, simulations of the turbulent premixed stratified burner operated with a hydrogen methane blend are performed. The validation of the modeling framework is performed by comparing the simulation results to extensive experimental data, allowing an in-depth analysis of the macro- and microstructure of the flame.