I think it's to give PCs a couple different approaches to overcoming certain obstacles in different situations.
The first way (with AC, hp and hardness) is to "attack" the object, which usually means damage it with a weapon or tool. This sort of situation is one in which you can hack at it several times (as chopping door with an axe, chipping at mortared bricks with a nail, etc). You might also use this method when trying to damage an object out of your control (as to sunder a weapon, or smash a wagon wheel on a passing carriage).
The second way is to beat the break DC: the "brute strength" method. This is usually for situations in which a character can hold/touch the object with no trouble and just wants to muscle it open/apart/etc with a lot of grunting or a mighty "hi YAH!". This sort of thing is for dramatic martial arts displays like breaking stones with the fist, as well as for "Conan the Barbarian" things like punching a door open, biting a discarded spear in half, pulling a treasure chest open, or crushing a beer can on one's forehead.
Some situations might be amenable to either approach. Breaking out of a jail cell window could be done by the rogue by "attacking" the bars with a makeshift saw; or by allowing the orc barbarian to simply grab the bars and rip the window out by beating its break DC with a strength check.
I'm not sure how it works in PF, though.
