The article for bulletproof vests in Wikipedia is in dire need of citations, but any site more aimed at explaining or selling such vests will confirm this:
Vests designed for bullets offer less protection against blows from sharp implements, such as knives, arrows or ice picks, or from bullets manufactured with hardened materials, e.g., those containing a steel core instead of lead. This is because the impact force of these objects stays concentrated in a relatively small area, allowing them a better likelihood of puncturing the fiber layers of most bullet-resistant fabrics used in soft armor. By contrast, stab vests provide better protection against sharp implements, but are generally less effective against bullets.
The reason stab vests are not great against bullets is because they handle impact differently. The bullet may not pierce you, but you still take the full impact at the contact point. Bulletproof vests on the other hand redistribute impact.
There is still the matter of arrows, which aren't properly stopped by either. You may use some handwavium for those.