Our nails are actually a specialized and reinforced form of epithelial tissue. When they mature, they lose their living property and become dead cells. Normally, there are no nerve cells within a mature dead cells and therefore, clipping of the nails do not produce any pain stimulation to the body. This goes the same to our hair.
When you observe any cross section of the skin, the receptors that branches from the nerve is only found within the skin. The hair are just outgrowths of the skin and there are no pain receptors connected to it. The hair roots are connected within the skin in a structure called hair follicle. Within this structure, the pain receptors is abundant. Which is why we feel pain when we pull them instead of cutting them.