The interaction between carbon buckminsterfullerene (C60) and carbon nanohorns (also referred to as nanocones) with different tip angles is investigated theoretically. Attachment of C60 to both the interior and the exterior of the horns are considered. Calculations cover a range of cone angles from flat graphene, through 114 degrees, 84 degrees, 60 degrees, 39 degrees and 20 degrees to fullerene pair interaction. Full DFT/LDA calculations are performed and the influence of dispersion forces are considered independently using a numerical potential. Fullerenes bind weakly to the external nanocone wall with approximately 2.9 angstroms spacing (0.5-0.9 eV binding energy), showing no discernable trend with cone tip angle. Fullerene binding inside cones is significantly stronger (> 3 eV), primarily due to strong dispersion force interactions, with higher (approximately 3.1 angstroms) fullerene-nanohorn spacing. In this case, the binding energy increases with number of pentagons in the tip. In all cases the fullerenes will be freely rotating below liquid nitrogen temperatures. For pristine cones and fullerenes, the fullerenes will experience a driving force towards (away) from the nanohorn tip when inside (outside) the nanohorn.