My initial objection is that nose filters air, mouth is for eating but is used for breathing also, plus they both are used to create sounds. What is the cause and reason in this case, why do we need two "holes" and one is not good enough?
Neither the nostrils nor the mouth originally evolved for breathing. Fish have (two pairs of) nostrils which they use to smell and mouths which they use to eat, but they breathe through their gills. Some lobe-finned fishes (the ancestors to tetrapods) evolved a connection between the posterior nostrils and the oral cavity called choanae. A fossil called Kenichthys is a transitional form in this development. The evolutionary reasons behind this development are not particularly well understood. (This should not be surprising when discussing something that happened in the Devonian. It's only recently that the discovery of Kenichthys ended the controversy of whether chonae in tetrapods are homologous to posterior nostrils in fish.)
Basal reptiles have nostrils and mouths and can breathe through both of them, but do not have a separation between the oral and nasal cavities. The next important development is a secondary palate (which separates the nasal and oral cavities). This allows animals to continue breathing while swallowing food. Animals without such a separation must hold their breath while swallowing. This ability is certainly useful in many situations, and unsurprisingly several solutions to this problem have evolved in different lineages.