I think a good study that shows this interesting contrast in Mediterranean/Helleno-Roman religion is Athens which had the sacrificial, calendrical polis cults that typified any significant community and also were the "keepers" of one of the oldest and famous Mystery Cults, that of Eleusis.

The system of Hellenistic sacrifice was based in the principle of mutual reception, relation and hospitality between Men and Gods, do ut des "I give so that you may give" . Greek sacrifice compared too some other cultures (Vedic comes first too mind) was relatively simple and communal affair, consisting of slaughter, butchering, hymns and burning. Here the Polis both reaffirmed the values, art, religious knowledge and history of the community, refreshed the beneficial mutual relation between the world of Man and Gods for both individuals and the polis and brought the community together for mutual expense (payment of sacrifial animals, plays etc), Grief and Joy.

But like many pre-Abrahamic Mediterranean religions, immortality was reserved for the gods and death was the lot of every Man save a heroic few. The importance of Household and family raising in part derives from this, bearing the next generation and passing on through Social sacrifice the things that made life beautiful and joyful too them was the closest thing in the traditional Greek religion too immortality. The afterlife was devoid with the Joy of life, dark and shade-like. Broadly, the religious "niche" of the Mystery Cults was it's promise of Immortality for those who have been iniated into them

The Mystery Cults were closed because they believed they held esoteric and difficult teachings that one has too be guided experientially through and which some were simply inelligbile too be taught. In the case of the Eleusinian Mysteries one had too be able to speak Greek and had never killed (presumably another person, as this would exclude a number of people).

Full iniation into the Mysteries took two years and lasted several days, beginning with Festival including purifying sea-bath, sacrifice, feasting, procession, ritual cursing, consuming ritual foods and all-night vigil.

Then at a specially made building the iniation began, consisting of ritual play, lecture and display of sacred objects. We are unaware of the exact nature of what was said, taught or spoken, but we have some idea from the philosophical ideas/writings of those who were initiated in them and the general direction of Hellenistic religious thought as the Mysteries were popularized. There was a wide variety of tales, ethics and practices of the Mystery Cults, but fundamentally as said earlier it taught experientially the Immortality of the human soul and it's nature too transmigrate, too die and rise as attested by the mythology the many mysteries were based upon.

For fun, because there is some doubt about it's authenticity, possibly from a copycat cult, but I think still it is haunting and mantra-like. The final, great and wonderous secret of the Mystery at Eleusis apparently was "An ear of Grain in silence reaped" as attested by Christian writer Hippolytus of Rome who himself was not an initiate.

Finally I think a good cap too this answer of the place in Hellenistic religious life and feeling that the Mysteryies held, Cicero writes in his Laws:

"For among the many excellent and indeed divine institutions which your Athens has brought forth and contributed to human life, none, in my opinion, is better than those mysteries. For by their means we have been brought out of our barbarous and savage mode of life and educated and refined to a state of civilization; and as the rites are called "initiations," so in very truth we have learned from them the beginnings of life, and have gained the power not only to live happily, but also to die with a better hope." Cicero, Laws II, xiv, 36

Sources:

Religion in Hellenistic Athens by Jon Mikalson

The Eleusinian mysteries & rites by Dudley Wright

The Mystery Religions by S. Angus