Coconut water is just coconut water. Coconut milk is either the blended coconut meat that's thick in a can (for cooking) or thinned down in cartons (for drinking) or especially thick in a bag in a box or something (for cooking also).
No one should be calling coconut water coconut milk.
As if it weren't confusing enough! :grr:
Coconut water=delicious
Coconut milk in foods=delicious
Coconut milk for drinking=
Where I am they market coconut water in both cans labeled "coconut milk" and "coconut water" and then make products using coconut water and call them "coconut milk" products, as well as marketing coconut "milk" made from modified coconut water.
Popular desserts often rely on something like dairy or eggs to bind it together, so it's harder, but not impossible to make them vegan.
Never tried this kind, but I love their chocolate chip cookie dough coconut ice cream.So Delicious makes a coconut milk based ice cream that has shredded coconut flakes in it which is amazing.