People often confuse these two terms and talk about "demand" when they really mean "quantity demanded." Here is the difference: demand (in economic terms) is the amount of a good or service that people are willing to buy at all possible prices. In other words, when you graph demand, demand is a line.
By contrast, quantity demanded is a point. It is the amount that people are willing to buy at any given price.
This matters because when economists talk about an increase in demand, for example, they mean that people will buy more at the same price than they would have before. By contrast, people will often say "the price went down so demand went up." This is economically inaccurate because a change in the price of a good or service only affects the quantity demanded, not the demand.