Aetiology of the word "fullerene" in Chemistry
Fullerenes are distinct (meaning every fullerene type has a well defined structure) carbon-only molecules, where all atoms have three bonds ($sp^2$, or nearly $sp^2$, as they're not exactly planar).

Leaving out very freakish topologies (which are unlikely to be stable and not fullerenes), this leads to a closed shell-like structure where all surface elements are pentagons or hexagons. (Which are not perfect, bond lengths differ slightly if the edge is shared by two pentagons, two hexagons, or one-and-one.)

The simpler, smaller forms are spherical (C60) or prolate (C70, ...), larger ones can probably have a surface that is not completely convex. (anyone seen oblate fullerene structures?)

A carbon nanotube with a well defined, closed end structure can be a fullerene, but they usually haven't got one.