Broth (whether vegetable or meat) is thinner -- it makes a good soup, but unless you thicken it with flour or cornstarch, it won't produce that thicker texture that characterizes curries. Broths also don't have the high fat content of coconut milk. Because of these factors, coconut milk produces a different mouth-feel than broth does.
I suspect, but do not know, that coconut milk also acts as an offset to the stronger spices in most curries, like yogurt does in other spicy dishes. (Not all curries are spicy, I know.)
Coconut milk, unlike animal milk or yogurt, also does not require refrigeration.