To me 'significant impact' would mean something along the lines of Guild Wars 2 where your personal story literally shapes the game you are playing. It gives you your motives and your 'whys' for who you are and what you do.
The Origins in DA: O were good examples as well because you had a reason as to what leads to each point. You've obviously had combat training in the Human Noble/Dwarf Noble Origin (favoring Warrior class), you had to be tough/street smart to survive in Dwarf Commoner (making either Warriors or Rogues viable), as a Dalish Elf you had to be a strong-arm of your tribe (I lean towards Rogue for the Ranger spec but either does), as a City Elf you've relied on your wits and know what it's like to survive (leaning towards Rogue), and with the Mage, well that doesn't need explanation
As much as I liked DA2 I will admit one of DA2's weakest points was not being able to play that first year as a mercenary or smuggler. In fact I don't think it even made a difference at all besides who you dealt with since it doesn't seen to impact anything in the game at all save for a minor sidequest. That was just plain bad. I also feel since Hawke was a more predefined character (you're the default in the intro and probably DA3 if he/she shows up), it was more about making choices with a mold-able putty crossed with a fixed doll (you get to define Hawke as far as his/her personality, romances and final allegiances go, but as far as his/her ultimate fate, being a noble, being the go-to man/woman for everything ala Shepard, it also made Hawke pretty much fixed).
Knowing these backgrounds will not be playable has me thinking more along the lines of Mass Effect, where it colors your background but doesn't ultimately add up (no matter what, Shepard is still an Alliance soldier regardless of how or even what class he or she is). I hope it'll be far more than DA2 and hopefully more than Mass Effect.
I love multiple playthroughs and experiencing different kinds of characters, and what better than to have characters that come from divergent backgrounds. I'd love to be able to take one character who was raised as a strong-arm of the Chantry, devout in the Maker's ways and Andraste's words, and be able to hammer it through people I come across and see at any cost. I'd also like to be able to take a character who was dirt-poor, unaccustomed to the noble ways, and perhaps it shows in some quests throughout the game involving royalty or nobility (having no manners, not knowing what to say). I don't like the idea of my character already being omnipotent or already knowing what to do. A raised on the streets rogue who knows the way of the underworld is going to be able to bribe, threaten, and navigate the seedier parts of town while a noble warrior is likely to believe a shady person, get robbed, yet unlike the rogue, the noble warrior may know how to effectively deal with nobility and even be able to woo or charm that person in order to get ahead. A mage...well only a mage should know about magic and dealing with the Fade, and since there probably will be a Fade sequence, they should have an easier time navigating. I want to see disadvantages as well as advantages. Just like I want my character to ONLY be his or her class, not Shepard/go-to person/can do anything and everything even if it doesn't fit the role/outside his or her class and abilities.
Backgrounds I'd like to see (feel free to disagree, just ideas):
Warrior: Chantry Templar (good with fellow Templars/Chantry dealings, Noble Chevalier (influence with royalty/nobility, higher wealth)
Rogue: Noble Bard (adept at persuasion and getting intel), Thug (influence with lower classes/bandits, bad reputation amongst people)
Mage: Apostate (bad with the Chantry, at risk of being hunted by Templars, unlocks unique magics), Circle Mage (okay with Templars, trained in navigating the Fade)