What is the difference between capillarity and osmosis?

The basic difference is capillary action draws liquid through a passage based on pressure differences, therefore able to work against gravity as will always move from the source of liquid regardless of the direction.

Osmosis is travelling through a membrane from an area of high concentration of a dissolved compound to a lower one. Therefore one involves passage of any liquid by capillary action, while the other is about the passage of what the liquid is carrying as a solvent.

https://en.wikipedia.org/wiki/Osmosis

What they have in common is they are both spontaneous movements but one is the liquid itself while the other is within the liquid.

Osmosis is the movement of molecules of the liquid in which a substance is dissolved (the solvent) from one place to another through a semi-permeable membrane. This movement tends to go from a region with a low concentration of the solute to one where there is a higher concentration, hence equalising the concentrations. The membrane is called semi-permeable because it allows the solvent molecules to pass through, but not the dissolved particles.

https://en.wikipedia.org/wiki/Osmosis

Capillary action is movement within a space, usually a narrow tube, caused by interaction between the molecules of the liquid and those of the surrounding material. The interaction of the surface tension produced by forces within the liquid and adhesive forces between the liquid and the surrounding material pull the liquid along the passage. It is most commonly talked about where this movement is against gravity, although it operates under other situations, too.

https://en.wikipedia.org/wiki/Capillary_action

So osmosis is movement of a liquid from one place to another physically separate one; capillary action moves a liquid along with anything that is dissolved in it from one lace to another area to which it is directly joined.