Final answer:
The mystery cults that affected Rome included those of Mithras and Isis, known for secret initiatory practices. Vesta, on the other hand, was not a mystery cult; it was a state-sponsored cult without the esoteric initiation practices.
Step-by-step explanation:
Mystery cults were religious groups in the Roman Empire, focusing on secret initiation rituals dedicated to a deity. Among the mystery cults that affected Rome were those dedicated to Mithras and Isis, known for their esoteric and initiatory characteristics. These mystery cults were characterized by a hierarchical structure of initiation, with particular beliefs and practices, and they stood apart from traditional Roman religion.
The cult of Mithras, in particular, found popularity among Roman soldiers and originated in ancient Persia, emphasizing secret worship rituals around the sun god Mithras. Isis was worshipped through rituals that likely combined elements from Egyptian tradition and adaptations made for the Greco-Roman world. However, one cult listed in the question, Vesta, was not a mystery cult affecting Rome in the way that the cults of Isis and Mithras did. The worship of Vesta was a state-sponsored cult, focusing on the hearth and home, and did not involve the secret initiatory practices that were characteristic of mystery cults.