Unfortunately this is quite a difficult question to answer. However, seeing as I am the only flaired user in a related area, I'll try to respond to it as well as I can.

Let me start by stating that I am not actually going to answer the question as asked. The idea of comparing such vastly different religious traditions as the greco-roman and christian ones is by itself difficult. The Puebloan religous traditions are themselves a world apart from either of these, having formed on a separate contact with no cultural connection. To put it simply, comparing these religions is at best problematic and at worst impossible. To make matters worse, none of these religions are really a singular, cohesive tradition. Within each is a number of distinct traditions that take subtly different forms and offer their own perspectives on the world. And to compound that issue, none of these traditions were static. They shifted and warped over time as they interacted with other traditions and people.

We can discuss this subject if a few modifications are conceded though. First, let us restrict our discussion to a specific group and period, say the village of Oraibi during the early 20th century. Puebloan religion varied from people to people and even from village to village or clan to clan. The Hopi of Oraibi is well-documented and had been a tremendously influential village until the late 18th century, so many features of Oraibi religion are present in other Puebloan groups. Not only that, but this particular village was the host of one H.R. Voth, a mennonite missionary who provided a thorough study of Oraibi religion from a Christian perspective. We needn't worry overly much about the tremendous time period between the 16th and 20th centuries either. As Geertz^1 and Spicer^2 state, the traditions changed little even through the christianization attempts of the Catholic Spanish, the Mormon settlers, and the Mennonites missionaries. Indeed, as Thompson and Joseph^3 point out, Oraibi religion provides a set of social institutions and incentives that discourage radical change. It need also be noted that the mennonite mission under H.R. Voth was essentially the first group to understand the Hopi religion for the purposes of christianization and this provides us with valuable information for a comparative understanding.^3

The question I am instead answering is related, but not identical to your own.

What are the major facets of religion in Oraibi that might seem unusual from a christian perspective?

It's perhaps best to begin by illustrating the general framework of Oraibi mythology. In the beginning (note that the hopi conception of time is very different from the western), the universe was created. Most conceptions of the creation story continue with the first two worlds being destroyed and humans coming to the third world. In the third world, the man that would become the leader of the bear clan became distressed of this world and went above to the darkness of the fourth world to find a new home. There he met with Masauwu, who bade him to return with his people. When he did, a group that would become the spider clan snuck out of the third world behind them. Depending on the clan and the village, many hopi stories will diverge from this point onwards. Generally, each of the clans that would later settle in Oraibi had a period of wandering. The parrot clan wandered from the far southeast of modern Mexico / Southern Arizona. The spider clan moved to Keams canyon and was forced to move by drought. From the east came the Snake clan and from the South came the bear clan, the first to settle at Oraibi. It was there that either Masauwu or Soqomhonaw gave to the chief of the Bear clan the rights to the land, admonishing him with a warning that the rights to the land were a permanently temporary loan. If the hopi should fail to properly honor their gods, he would revoke them.

After spirits (the proper term 'hikwsi' does not translate well, but this is perhaps the most near concept) die, they may go to two places. Those humans that have not been initiated into the world by the proper rituals are bound to their familial homes. Those that have been initiated return to the third world, within which is contained the skeleton house at the bottom of a great canyon. The third world is perhaps superficially similar to Greek underworld. The skeletons there eat only the spirit of food and drink and throw the rest away. In this afterlife the clan system is preserved and each spirit shares his ancestral clan. Those who have been abusive to women or children are forced to drag large stones or thorny cacti for a time before they may join the asphodelian spirits of their ancestors.

It's at this point that I am struggling to relate anything further to christian theology. It's not really possible to discuss Oraibi religion much further without a discussion of the kachina cult. Perhaps most nearly, it's roughly an animistic belief system that everything possesses an essence. The kachinas themselves are generally called spiritual incarnations of these things, but this description is incomplete. Most commonly among Oraibi, kachinas are seen as spirits of the lower world. Not deities in and of themselves (as H.R. Voth mistakenly calls them), but with the ability to affect the environment. The kachina cult seeks to influence the kachina through ceremony and ritual figures, which are imbued with a special relationship to their respective kachinas. However, the traditions take on a more interesting role than simply animism. They inform the oraibi way of interacting with the environment, including their social structures. Entire books have been written on the subject, which is integral to Hopi society. The most accessible one (and heavily, heavily flawed) is The Traditions of the Hopi by H.R. Voth. As I mentioned earlier, Voth approaches Oraibi traditions from a mennonite perspective with the intent to christianize the natives. He was one of the first outsiders granted deep access to Oraibi and his actions made him perhaps the most hated person within the village even today (although only the elders remember his name specifically). Accordingly, he's also the last reasonably comprehensive source available.

^1 Armin W. Geertz Book of the Hopi: The Hopi's Book?, Anthropos (1983), pp. 547-556 Anthropos Institute

^2 Spicer, E.H. Cycles of Conquest. The Impact of Spain, Mexico, and the United States on the Indians of the Southwest, 1533-1960, (1976), The University of Arizona Press.

^3 Laura Thompson and Alice Joseph White Pressures on Indian Personality and Culture American Journal of Sociology, Vol. 53, No. 1 (1947), pp. 17-22 The University of Chicago Press

^4 R. Forrest, Earle Missions and pueblos of the old Southwest; their myths, legends, fiestas, and ceremonies, with some accounts of the Indian tribes and their dances; and of the penitente (1929), pp. approx. 27 Rio Grande Press