What is the difference between conkers and chestnuts?

No. Sweet Chestnut is contained in a capsule with soft spines and roasted ,

Horse Chestnut is contained in a spherical capsule with sharp spikes and is inedible.

Latin genus,Castana hence 'castanets'. To be absolutely correct a roasting chestnut is the seed of Castanea sativa the Spanish Chestnut. Conkers are the seed of 'Aesculus Hippocastanum the Horse Chestnut.