Technically, a Flak Jacket is Body Armor, however these "Jackets" were designed to protect the wearer from LOW Velocity projectiles... think flying debris and shell casings.
Sadly I have witnessed people selling Flak Jackets as "Bullet Proof" ... the problem lies with the velosity. Bullets by their very nature move at a high velocity. While it's true that some Flak Jackets have been demonstrated to defeat a bullet, this ability varies widely and is in direct correlation to the velocity and angle of the projectile, the distance it was fired and the weapon it was fired from.
It was claimed that the M-1951 flak jacket could stop a 90 grain 7.62×25mm Tokarev pistol round at the muzzle of the gun. However, even the Vietnam era revised flak jacket was not really designed to stop an AK-47 round (7.62×39mm) fired at close range. Nevertheless, it did a good job of stopping shell blasts, rocket fragments, and slow speed bullets fired at a moderate range.
Flak Jackets have their place and purpose HOWEVER, they are not "bullet proof"