I’ll let Ian expand on his. I describe myself as a Classical Hellenic Neopagan (how’s that for concise?

). I follow the Greek pantheon, as they have come to us through the history of Western civilization. The framework in which I do this is as part of a group called Ar nDraiocht Fein, describing itself as a Druidic group, but actually involving, as I believe Ian said, reviving the polytheistic practices of our Indo-European ancestors within a modern context rather than only Celtic ones. Not all who follow the Hellenic gods would have the same practices that I do.
We look to the best available information from modern sources about ancient practices (archaeology, anthropology, etc) to learn more about how the gods interacted with our ancestors, but personal experience is also key. I am not an ancient Athenian, I don’t live in that culture, share that worldview, etc–my experiences of the gods may well be somewhat different than that Athenian’s was. It’s somewhat akin to a modern Christian looking for the historical Jesus. The information available may be useful to you in understanding the social and political reality that shaped the direction of the early Church, but it is your relationship with God here and now, in your personal life, and in the religion as it is practiced now in our society, that is likely of primary importance to you spiritually. If the only connection that existed was academic, I doubt it would sustain you for very long.
Knowing through the stories the offerings that the gods found pleasing from our ancestors and the ways in which the gods chose to interact with them can give us a starting point to see how we might be able to develop a relationship and call upon them in our modern society. Historically, offerings (sacrifices) of food, jewelry, art, poetry, song, dance, libations of oil, wine, honey, milk were all welcomed by the gods, and are more in keeping with modern living situations, laws and mores (hard to keep a hecatomb of oxen in the usual suburban backyard–you need a heck of a pooper scooper

). The relationship to the gods is as a host to an honored guest, one who is more powerful, rather than master/slave or parent/child.
For me, the gods that call me are the Hellenic ones. For others in the organization, they may hear the call of the Celtic, Baltic, Roman, Norse, Germanic, etc. When I say sect, I mean that our pantheons are different and so we are part of a special interest group within the organization that follows the Hellenic gods. That is the source of the comment about different gods, different desires for worship.
As Ian also said, it is about orthopraxy (right action) rather than orthodoxy (right belief), so the similarities would be with ritual forms, liturgy, ethical guidelines rather than creeds–no blood sacrifice (human or animal) regardless of how one’s gods were worshipped in antiquity, no illegal substances as part of rituals, no membership in racial supremacist groups, no discrimination along racial/ethnic/national origin/gender/disability/sexual orientation lines, etc.