My understanding is that partition chromatography is, for example, paper chromatography in which the analyte is partitioned between the developing, mobile phase and the water (naturally present from the atmosphere) in the cellulose matrix. The cellulose (paper) is there only as a physical support for the water.
Silica gel chromatography is adsorption chromatography---the analyte is adsorbed/desorbed onto/from the silica gel repeatedly.
However, I would like to hear what a modern text on chromatography has to say.