Basically, both of the verbs mean not allow doing something. The main difference between them is that while to prohibit means to forbid, to disallow, to proscribe, to ban, besides its first meaning, means to summon, to call out. By definition, to ban means to refuse to allow doing something. Besides, it also means to excommunicate someone from the Church. To prohibit means to forbid something by law or by the authorities. For example, cars can be banned out from a city center, and smoking can be prohibited in restaurants. As a noun, it was the legal act of not allowing drinking alcohol in the United States between 1920 and 1933. If we talk about this era, the notion is written with a capital letter.
