I will answer what I know and leave it to others to add more.
Two major changes that take place as cheese ages are fermentation and drying. In some cheeses time is also needed for mold growth such as Brie or Blue cheeses.
The fermentation is usually the more desirable. Bacteria is introduced when the curd is formed. The bacteria consumes lactose and turns it into lactic acid and other flavorful byproducts. This acidifies the milk and is an important factor in curdling it and creating the cheese curd.
When the curd is pressed, excess water is removed. In the case of Parmesan most of the water is removed. This retards the progress of the bacteria. As the cheese ages the bacteria stays active and makes a lot of flavor. Think of a bread- a little yeast with a lot of rise time tastes much better than a lot of yeast and a short rise.
The acidity and flavor increases. Drying also plays a part in concentrating flavor.
Both the acidity and drying will change the protein structure of the cheese and is what gives Parmesan its characteristic crumbly texture. I can't speak to what these changes are on a molecular level.
Is there a food and/or dairy chemist in the house?