Capillary action is a surface effect, where liquid is drawn towards a surface that it is attracted to. Water in a drinking glass glass is drawn up at the edges because it is attracted to the glass surface. You can get capillary action with finely divided material like a candle wick. The wicking effect is melted wax drawn up between the fibres of the wick. In a wicking pot water is drawn up by capillary action. The liquid does not have to be drawn upwards but it is more noticeable and more useful in some ways if going against gravity.
Osmosis is where there are liquids with different strength of dissolved substance on either side of a membrane that the liquid can pass through. So water passes through the cell walls (the membrane) of living things from lower to higher concentration of dissolved substances if the conditions are right. Such membranes are usually very thin unlike an unglazed pot. Unless coated with something special an unglazed pot is not a semi-permeable membrane for the purpose, so water passes through under capillary action and gravity.
There are some references on the web like this one that are just wrong.