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There's no solid answer.

They are both "darker sides of the Doctor", but one is a physical manifestation of the Doctor taken from between his Twelfth and final incarnation, while the other is a dream manifestation of his unconscious mind. (Also, the Dream Lord seems to be more specifically the darker side of the Eleventh Doctor, while the Valeyard is the darker side of the Doctor as a whole.)

So, the best you can say is probably that they're both different characters, but both represent the same thing in similar ways.