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The process that primarily occurs within solids and relies on temperature differences is conduction. In conduction, heat energy is transferred through a material without the movement of the material itself. This transfer occurs at the molecular level, where faster-moving (and thus hotter) molecules collide with slower-moving (and cooler) molecules, allowing heat to move from the high-temperature region to the low-temperature region.

Conduction is particularly effective in solids due to their closely packed molecules, which can efficiently transfer energy through direct contact. As temperature differences increase, the rate of heat transfer by conduction also increases, making it highly dependent on these differences.

In contrast, convection involves the movement of fluids (liquids or gases) and occurs due to the buoyancy effects of temperature differences. Radiation is the transfer of heat through electromagnetic waves and does not necessarily require a medium (solid, liquid, or gas). Diffusion refers to the process of particles spreading from areas of high concentration to areas of low concentration, which does not specifically relate to temperature differences in the same way that conduction does. Thus, conduction is the correct answer for the process that occurs in solids and is influenced by temperature variations.