First you might want to check out my thread here:
http://www.ar15.com/forums/topic.html?b=6&f=46&t=280325
Second, let me comment on a few things:
1) The Amsec BF series is steel on the outside AND on the inside with the poured layers of fire insulation between the 2 layers of steel. I assumed that other safes were also steel on the inside and was surprised to find out that not all of them are (even a $2000+ Liberty Lincoln). Check it out and be certain.
2) Another thing I was surprised to learn about the Amsec BF is that the bolts on the hinge side are fixed. After thinking about it, I realize that this is probably and advantage more than a disadvantage. Why? Because (a) there's no way for the bolts to be pushed in if they're fixed in the open position and (b) because having less moving parts means less likely for a mechanical failure of the locking mechanism. The bolts on the Amsec are also thicker at 1.5" vs 1.25" on the Browning you referenced. The only thing that I think Amsec could have done better on was adding a bolt or two to the top and bottom of the door, which leads me to point #3.
3) As noted, the door has a 1/2" steel plate on the front, followed by layers of fire insulation and additional steel. This is one serious door for an RSC and you can feel it by it's weight. Some have claimed that the BF doors are 'pry proof'. I don't know about that but I'm guessing that they'll stand up very well.
All this said, the Browning safe looks to be good. I certainly wouldn't lose any sleep having one of them protecting my stuff. But given the (little) information I already know, I'd probably pick the Amsec BF, but you can call me biased because that's exactly what I already did.