This is a problem with attributes, specifically generic.attackDamage.

When putting the attack damage attribute on a helmet, it dealt increased damage when I punched a spider. When I switched to my sword and back to my fist, the spider took only normal damage. The damage did not increase when I used both the sword and helmet, only the sword dealt damage.

Helmet:

/give @p minecraft:iron_helmet 1 0 {AttributeModifiers:[{AttributeName:"generic.maxHealth",Name:"generic.maxHealth",Amount:40,Operation:0,UUIDLeast:852852,UUIDMost:258},{AttributeName:"generic.attackDamage",Name:"generic.attackDamage",Amount:10,Operation:0,UUIDLeast:852852,UUIDMost:258}],display:{Name:"Warrior Helmet"}}

Sword:

/give @p minecraft:iron_sword 1 0 {AttributeModifiers:[{AttributeName:"generic.attackDamage",Name:"generic.attackDamage",Amount:10,Operation:0,UUIDLeast:852852,UUIDMost:258}],display:{Name:"Warrior Sword"}}