The wild amaranth is Amaranthus Palmeri and was once grown by American Indians for the seed. In high nitrogen soils it can pick up an excess of nitrates and be mildly poisonous to livestock. The one best grown in the garden is Amaranthus Caudatus and the leaves and seeds are edible by both livestock and humans. The flower garden form of Amaranthus Caudatus has a shorter stem and longer flowers that often trail onto the ground and bears the old fashioned nam,"Love Lies Bleeding".
As an aside the Amaranthus Palmeri is one of the weeds that has learned to tolerate glysophosphate and is very worrrisome to Monsanto, which may mean it has picked up some GMO genes.