Inverse demand maps from quantity to value. It is useful to identify how much some units of the good are worth to the consumers. The central welfare concepts of consumer's and producer's surpluses rely on inverse functions in their definitions.
In addition there are special cases (e.g. perfectly inelastic demand) where defining inverse demand is just easier.
In a competitive market, price and quantity are simultaneously determined by the intersection of supply and demand. So neither the ordinary market demand function (quantity as a function of price) nor the inverse market demand function (price as a function of quantity) can be considered more significant or fundamental, or as expressing a causal relation in a more natural way.