What's the difference between Attack modifier and Damage Bonus?

Attack modifier adds to your chance of hitting.

Damage bonus adds to your damage.

Every time your characters attack, a number between 1 and 20 is randomly generated, and then your attack bonus is added to it. If that number is equal to or greater than the Defense stat of whoever is being attacked, you've hit, and another "die" (a holdover from the D&D roots of the system the game uses) is rolled (what size the die is depends on the weapon, the more powerful the weapon the more and bigger dice are rolled), and then various bonuses - based on the character's strength, as well as on what upgrades the weapon has, and what Feats the character is using - are added on to it to determine the total number of vitality points the opponent loses.

 

Generally, unless your damage bonus is really piddly, this means you're better off increasing your chance to hit rather than damage - you'll cause more harm if you hit with every blow for less damage, than if you hit only once every second or third swing for twice as much damage per blow. Of course, ideally you want to increase both...