I shop at all 3, however most frequently at the Gap. BR's items tend to have a more refined, upscale look. More semi formal to formal items as well. Overall quality may not be better per se, but similar items have more/nicer little details to them to go with the "look".
For example. I have a Camel colored wool car coat I purchased from the Gap. Nice clean look to it, nicely lined througout. It's definitely a coat I can wear with a shirt/tie/slacks or just a shirt/jeans.
BR also had a Camel coat of a similar style. It however was a wool/cashmere blend I believe. It had a much more tailored cut and fit to it, and wasn't nearly as bulky. While you could still pull it off with jeans, it was definitely dressier. Some nice details on it were little flaps it had that went over the outside pockets, a more suit-like lapel, and what I thought was nice was some detail stiching around the edge of the collar/lapel.
Old Navy didn't even carry a similar coat.
The coat I got was stickered @ $168. The BR version was $298.
I would've gone for the BR version in a heartbeat if 2 things happened:
1. I actually fit BR stuff well. As it is, due to their more "athletic" cut, I fall between a L and XL. I don't like the way either looks on me. 1 is too snug/tight, the other fits better but is usually too long (shirts). don't get me started on pants.
2. I could get it on sale. I rarely buy anything for full price from any of the 3 stores. My coat was at full price, but I had a $25 gift card, and got 20% off and double points for opening a Gap card account. So it was discounted quite a bit in that sense. I did see it on sale for around $60 tho a couple months later.
ON is great for the kids though. Good, cheap, and always looks good. i go there for stuff like tshirts and shorts.