The Bradley is actually made of aluminum, though ever since the US ditched the "must be able to swim" requirement they're covered in steel applique armor. BMPs have some advantages, primarily in profile (i.e., physically smaller therefore harder to see). From the front, however, the Bradley is resistant to the modern BMPs' main weapons, the same cannot be said for the BMPs' resistance to the Bradley's main weapons. Should a Bradley get hit and penetrated and the ammunition (which the Bradley has more of than a BMP, assuming only authorized stowage is being used) go off, the vehicle would be totaled same as any other, but no vehicle is ever going to be so thoroughly destroyed as to leave nothing but an unidentifiable puddle. You'd know what it had been, it'd just be completely unsalvageable.
BMPs are certainly worse off than Bradleys in the survivability department-they're lower to the ground, which means when a mine or IED goes off its energy is not dispersed into the air gap before hitting the vehicle's hull, they're less heavily armored in general, and the guys inside are extremely close together meaning that any penetrating hit from any weapon is going to cause more casualties. And getting out of a BMP is not a fast process because it's so cramped, so if one starts burning a lot of the guys inside are going to end up dead or burned as they bail.
Also the Bradley just has like three times the leg room for soldiers inside compared to the BMP. You can't win wars with leg cramps, y'know?