Through magnetic linear dichroism spectroscopy, the magnetic susceptibility anisotropy of metallic single-walled carbon nanotubes has been extracted and found to be 2-4 times greater than values for semiconducting nanotubes. This large anisotropy can be understood in terms of large orbital paramagnetism of metallic nanotubes arising from the Aharonov-Bohm-phase-induced gap opening in a parallel field, and our calculations quantitatively reproduce these results. We also compare our values with previous work for semiconducting nanotubes, which confirm that the magnetic susceptibility anisotropy does not increase linearly with the diameter for small-diameter nanotubes.