Bonus damage is what we call elenore. It adds extra line of damage per bonus damage modifier you equip. If you have 2 items with bonus damage, you'll see 3 lines of damage, 1 big and 2 smaller ones. Additional damage is what we call smash, there are 2 types, normal and critical. You can only wear 1 of each, 1 additional damage and 1 additional critical damage. They don't stack and there's usually a warning in the item's description. Only the highest of smash and crit smash will take effect. This does not add a new line of damage but instead it's part of the main damage number (the big one). I'm not sure about attack damage though. For any class, including summon-type in the future, these attack modifiers are good. You'll only have to worry about smash/crit smash. There's also the difference in application. Some smash, crit smash and even elenore has specific ways of application. You'll see "additional damage on bleeding", you'll have to keep an eye on those and must consider if your character can do that requirement. For elenore, the most powerful kind is the elemental kind, "Bonus [element] damage". This kind double dips elemental bonus. The thing you have to watch about this is your own skill's default element. If you're locked to a specific element, it might be hard to take full effect of this type of elenore. Ideally, you'll want 1 crit smash, 1 smash, as many elenore and other stackable modifier.