DR/Adamantine vs. Hardness

[Jon_Dahl]

What difference is there between hardness (animated object, caryatid column) and Damage Reduction/Adamantine (iron golem, flesh golem)?

[Karnith]

Probably the most important bit is that hardness applies to damage generally (e.g. weapon damage, energy damage, damage from spells, etc. though there are a number of things that bypass hardness), whereas damage reduction only applies to weapon damage (and a few other things, like some psionic powers). Per the SRD entry for hardness (http://www.d20srd.org/srd/exploration.htm#hardness):

Each object has hardness—a number that represents how well it resists damage. Whenever an object takes damage, subtract its hardness from the damage. Only damage in excess of its hardness is deducted from the object’s hit points (see Table: Common Armor, Weapon, and Shield Hardness and Hit Points; Table: Substance Hardness and Hit Points; and Table: Object Hardness and Hit Points).
Per the SRD entry for Damage Reduction (http://www.d20srd.org/srd/specialAbilities.htm#damageReduction):

A creature with this special quality ignores damage from most weapons and natural attacks. Wounds heal immediately, or the weapon bounces off harmlessly (in either case, the opponent knows the attack was ineffective). The creature takes normal damage from energy attacks (even nonmagical ones), spells, spell-like abilities, and supernatural abilities. A certain kind of weapon can sometimes damage the creature normally, as noted below.

[Anxe]

Hardness has all sorts of exceptions based on what type of object is being attacked as well. That's mostly open to DM interpretation, but an easy example is that wood does not apply hardness to axe attacks or fire damage.

[HunterOfJello]

If you're curious about how adamantine weapons or the substance in general works, then you can find the specific info in the DMG on page 283.

An adamantine weapon ignores all of the damage reduction that a creature with DR/Adamantine has. It also ignores the hardness of weapons and objects if they have less than 20 hardness.

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It's also worth noting that even though hardness allows objects to ignore other types of damage, there are specific restrictions on how different elemental damage effects objects. This is explained more in-depth on PHB page 165. Essentially, Acid and Sonic do normal damage, electricity and fire do half, and cold does 1/4th damage to objects (before hardness is applied).