The main difference between a GHMM and a LSD-DHMM lies in their output models in that the output model of a LSD-DHMM directly captures the context dependence between successive observations in determining the "hidden" states while the output model of the GHMM fails to do so.
the output model of a LSD-DHMM overcomes the strong context independent assumption in the GHMM and becomes observation context dependent.