Bullet proof vests are not worn as standard in military operation, they are used when they will be most effective: often in anticipation of the type of ordinance likely to be used against them.
They are bulky and cumbersome, and due to the fact they aren't very breathable, they leave soldiers prone to overheating. This still from the movie shows that they are wearing webbing, but seemingly not kevlar.
When this image is compared to a soldier wearing actual body armor, the differences are immediate.
That second photograph isn't strictly military issue, but it still serves the point. Look how much bulkier it is, and how much more restrictive it would be.