Scanning tunneling spectroscopy allows us to study the changes in the quasiparticle local density of states due to the presence of vortex cores in Bi2Sr2CaCu2O8+δ (BSCCO). The vortex core spectra are characterized by a gap-like structure similar to the pseudogap measured above Tc. We also observe small spectral structures inside the gap in the vortex cores. The observed low-energy structures are weaker than what one expects for localized quasiparticle excitations.