One of the most interesting attributes of the fullerenes (or whatever of the many names this allotrope of carbon goes by that you happen to prefer), is the strange way in which they become superconductors when extremely cold, whereas graphite does not.
As The Alchemist noted above, buckyball is very similar to graphite, in that its bonding is primarily sp2 hybridized, with each carbon atom forming 3 bonds, and sharing a fourth bond aromatically with its neighbors through pi bonding. The really glaring major difference between graphite and C60 is that graphite is flat, with "even" pi bonds among p orbitals of the atoms, whereas in buckyball you see a "bent" effect. Largely this has caused a great deal of speculation among organic chemists as to whether this "bending" of these orbitals is what causes the superconducting effect, and has led to some research into conical shaped carbon structures as a result.