How do hard cheese and soft cheese differ?

Soft cheese is unripened cheese made by coagulating milk proteins--or casein--with acid. Examples include cream and cottage cheeses. Hard cheeses are ripened--or aged--cheeses made by coagulating milk proteins with enzymes (rennet) and culture acids. The cheeses are ripened by bacteria or mold. Cheddar, Swiss, Colby, brick and Parmesan are bacteria-ripened hard cheeses. Blue, Roquefort, Camembert and Brie are mold-ripened cheeses.
 
Because aging depletes cheese of moisture, hard cheese is drier -- and harder -- than soft cheese. Bacteria can't grow as readily in dry environments as in moist environments, which is why hard cheeses keep longer than soft cheeses.
 
However, differences among hard and soft cheeses may not always be clear cut, especially among semi-hard and semi-soft varieties. Additionally, some cheeses can be made in softer or firmer formats, depending on how long the cheese is aged and the pressure with which it is packed into molds.
 
Examples of soft, semi-soft, semi-hard, and hard cheeses are as follows:

• Soft:  These include Brie, Camembert, and Ricotta varieties.

• Semi-soft: These include Gouda, Havarti and Jack, all of which are suitable for melting or eating as is.

• Semi-hard: These include cheddar cheeses. Edam is a cheddar cheese. 

• Hard: These are better known as grating cheeses and include Parmesan, Mimolette, and Asiago. 

 
Source(s):
Wikipedia.org "Cheese"
Whole Foods Market "Cheese"