They exist to categorize weapons into groups and to provide boost to the encountered characters attack and defense numbers in combat, as some Enemy have traits that gives them an advantage over a specific weapon class.
For example, the Grove Sprite has the Floral-Aid trait that provides a 27% chance of +4 attack against weapons in the Strong weapon class.
Also, Weapon Class react to certain Enemy Classes. If you're using the wrong Weapon Class when encountering an enemy character, you will get penalized (your attack power will be reduced) by approximately 10%.
Weapons may have both a Class and a Power type. The effects of both stack, enabling weapons to be quite powerful in the right situations.