Darjeeling is more aromatic and (for a fermented black tea) tastes quite close to a green tea. Assam tastes stronger and is more astringent on the mouth when drunk without milk.

I always imagined that the difference in flavours reflected the different geographies - delicate growth in the Darjeeling hills versus robust growth in the warmer Assam plains. It might also be differences in the way the teas are fermented.