Yes, they're the same. But 'conkers' are from the 'Horse Chestnut' tree, and are very bitter, making them unpleasant to try to eat.
Edible chestnuts, like the ones we see in stores at Christmas, are from the 'Spanish Chestnut' tree.
Strangely, though the leaves and the chestnuts look in some ways similar, the two tree species are not related to each other.
They are not related at all and i personally wouldn't touch conkers with a barge pole
Horse chestnut (Aesculus Hippocastanum) are from the Hippocastanaceae family whilst sweet chestnuts (castanea sativa) is from the beech family or fagaceae family.