If the context is two people in a relationship, "need" is used to express very strong feelings about being with a person. Since we need water and food, saying that you "need" someone is like saying "I cannot live without you". The word "want", in a relationship setting usually refers to the desire for intimate relations with the other person (mostly sexual).

Need is something without which a person cannot survive. For exam, the human body "needs" water to survive Want is a desire which a person may or may not get. For example, I want to have ice cream