What is the difference between capillary action and transpiration?

Basically, transpiration is a means of absorption, capillary action is a means by which liquid flows.

Transpiration is when a plant absorbs water in its roots. noun. Transpiration is the process where plants absorb water through the roots and then give off water vapor through pores in their leaves. An example of transpiration is when a plant absorbs water in its roots.

http://www.yourdictionary.com/transpiration

Capillary action (sometimes capillarity, capillary motion, or wicking) is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity.

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

With transpiration the important element is the water (or gas, see second link) moving through and exiting the object that takes it in. The method of transport (of which capillary action is one) is not relevant. It just so happens that plants pretty much do it the same way in all their various structures, and in all the different kinds of plants, but if a plant sprouted arms and a mouth bent down and sucked up water in a straw, or built a motorized pump on it's stem, it would still be transpiration, botanically speaking, as long as some was released to the atmosphere

https://www.dummies.com/education/science/biology/how-plants-pull-and-transport-water/
http://www.biology-online.org/dictionary/Transpiration