A platform bulletproof vest is plate carrier that may have some anti-ballistic capabilities. Often, it will refer to a NIJ Level IIIa vest with pockets for hard plate armor.
Depending on the intended use, a bulletproof vest can be made from either soft anti-ballistic fabric, like Kevlar, or hard plates made from composite materials, like Silicone Carbide.
Generally, yes. It is legal for all law-abiding citizens to wear bulletproof vests both in public or private, provided that the property owner where you are allows for such behavior.