Having rings instead of a moon would affect the stability of the world's rotation. The moon acts to stabilize the degree of tilt of the world's axis, similar to if you've ever spun an object on a string around your hand. The weight of the object and pattern of its rotation keep your hand moving rhythmically. Without that, the world would wobble on its axis much more than it does with the moon. That would result in much less stable seasons, as the degree of tilt would vary wildly over time.
Furthermore, with astronomy, you can tell where you are on the world latitudinally (North and South). That's because you can look at the stars and judge their angle from where you started. The North star is always exactly north, so if it's halfway across the sky, then you're halfway across between the equator and the north pole. You can compare your latitude from when you left, to where you are at any point, and always know how to get back to the same latitude. However, this only works for us because our rotation is so stable. Without the moon, we wouldn't have that stability, and the north star would only be vaguely north. But we have the rings! Say those rings are on the equator. Now we don't need the north star, we've got these rings, which do the same job and more! Furthermore, those insane seasonal shifts could be predicted years in advance. By comparing the angle of the rings to the angles of the stars, people could tell how much the axis of the earth was shifting, and thereby predict (to some degree) the shift in climate.
Plus, with no moon, this world has no tides. No tides means much calmer seas, which means it would be much easier to set out at sea. There would still be storms, and currents, sometimes very strong currents, but it would be an order of magnitude safer to be on this world's seas than the seas on earth.
However, the stars alone never solved longitude. Just because you know how far you are north and south doesn't help with east and west. It seems simple enough though, doesn't it? The earth rotates around once every 24 hours, so just check every 24 hours, and compare where the stars are now from where they were when you left. That works perfectly in theory. But not so well in practice. It took until 1714 for a good enough clock to be developed. Until that point it was "dead reckoning" which basically meant that you just took as careful track of your location and time as you could, and hoped for the best. Many people died by messing up even a single calculation. But we have the rings! The rings in lower orbits would complete their orbits far quicker than rings in further orbits (as anybody who has tried to dock in KSP can surely tell you). Also, there's almost no way the rings would be perfectly without flaw or imperfection. Even a slight shift in color along the rings would be enough. People would be able to tell the time based on how far apart imperfections in inner and outer rings were from one another.
Add this all up and we can almost certainly expect far more nomadic civilizations. People would use their astronomical skills to predict the seasons, and gauge where the climate would be right for their food sources of choice. Food sources would not stay in one place, and neither would most people. In our world, people could generally get through life staying in one spot; building up reserves for winters. But if the winters would get progressively worse for 20 or so years, then it would make much more sense for many more people to get up and leave at the first sign of one of these mega-seasons. Sure, some people would stay, but most of civilization would hug closer to the equator and to shorelines. There would be much more interaction (both friendly and hostile) among cultures, as more people would be moving back and forth on land and sea, especially at sea. It's possible that in this world the ancient Greeks could get all the way to the Americas. It's just as possible that the natives would never have made it across the land bridge into the Americas, although the Polynesians would definitely make it.
I don't know nearly as much about religion and culture, so I'll leave it to others to fill in how those would be different.