Here, Buckminsterfullerene does not conduct electricity because they do not contain free electrons but graphite is the only carbon compound that being non-metallic in nature having a covalent bond but shows conductivity due to the presence of free electron.
Though the free electrons are present in the molecules of Buckminsterfullerene they are held by Vanderwall forces. Hence their movement is inhibited. As a result, Buckminsterfullerene doesn't conduct electricity.
Graphite is a soft, slippery, and black material that is an allotrope of carbon (three major allotropes of carbon - Diamond, graphite, and fullerene).
Each carbon atom in graphite is directly linked to only three carbon atoms through covalent bonds. Therefore, of the four valence electrons in a carbon atom, only three are used for bonding and the fourth is relatively free and can move from one carbon atom to another. This free electron in the molecule results in the conductivity of graphite.