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A hydrophobic core, like a hydrophilic core, is technically a characteristic of a secondary structure but is not a type of secondary structure itself. There are also examples, like amphipathic helices, that have the hydrophobic and hydrophilic residues partitioned on either side of the outer surface of the helix.
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Hydrophilic core and hydrophobic core are considered part of the protein molecule itself. Secondary structures can be alpha helices and beta sheets which make up the tertiary structure. The core of the protein molecule is usually made up of hydrophobic amino acids with the hydrophobic R groups facing inwards. In a sense, I think hydrophobic core is the characteristic of the proteins final folded tertiary structure.
Biochemist and Graduate student at Johns Hopkins University
hydrophobicity is just a measure of how soluble an amino acid is in water, respectively. In amino acids, hydrophobic molecules will be located on the interior portion of the protein, while hydrophilic will be more likely to be in contact with any aqueous solution.