No both red and white phosphorus have same symbol P because both are allotropes of phosphorus which means they have same atomic number,mass , chemical properties etc, but they differ in physical properties such as boiling temperature, melting temperature, solubility etc
White phosphorus is obtained by roasting Calcium phosphate rocks and then reducing them using silica
White phosphorus reacts with air vigorously that is it burns instantly in presence of oxygen and show phophorescence property ( it emits bright luminous green light in dark in prsence of small amount of oxygen ) , hence it is stable below 125'C and red phosphorus is stable above 125'C. If you heat white phosphorus upto 125'C in absence of air you get red phosphorus.
White phosphorus has 4 atoms of P linked in tetrahedral shape, where as red phosphorus have amorphous network arrangement hence red phosphorus is more stable.
White P is soluble in carbon di sulfide and small amount in water, but red P is not soluble in both water and carbon disulfide.
white P is used in explosives, match boxes, incendiaries, bullets etc
There are other allotrophes of Phosphorus look up in wikipedia with keyword "Allotrophes of Phosphorus", there are violet, black, di phosphorus etc
hope i was helpful