A change in supply and a change in quantity supplied are different things. The first is shown graphically as a movement of a supply curve while the second is shown as a movement along a curve. The first is caused by changes in costs and incentives that change how much a producer can and will produce at a given price. The second is caused simply by a change in the retail price of the product.
In economics, supply is defined as the amount of a good or service that producers are willing and able to sell at each possible price. This is depicted as a curve because there is a particular quantity (this is shown on the x-axis of the graph) that they will supply at each possible price (y-axis). In order for supply to change, something has to happen to make producers willing and able to produce more (or less) of their good or service at a given price. For example, if a supplier is making blue jeans and the price of cotton drops, they will make more profit off of each pair of jeans sold for a given price. Therefore, they will want to sell more at that price and supply will increase.
A change in quantity supplied is different. A change in quantity supplied happens when the retail price of the good or service changes. For example, if I am selling jeans for $35 per pair and all of a sudden jeans are seen as a fashionable item, people will want more of them. Demand will change, raising the price of jeans to, say, $40. At that point, I will produce more jeans because the law of supply says that, all other things being equal, producers will be willing and able to sell more of their product as the price of that product rises. I am not willing to produce more jeans at $35 per pair; that would be a change in supply. Instead, the price went up to $40 causing me to produce more jeans, but at a different price.
So, changes in supply make producers able and willing to sell more (or less) at a given price. By contrast, changes in quantity supplied are caused by changes in price, causing producers to sell more (or less) at a different price.