First, a quick disclaimer; this is mostly not about religion, anthropology, sociology, or economics. This is mostly about physics, and a little bit about how short-lived humans are.
Humans like taking a skill and doing it to death. This has got us a lot of really cool technology, but sometimes it gets stuck -- like when chipping stone points hit a plateau for about 100,000 years. Buildings are no exception, and we have spent a lot of time building dirt mounds in many interesting forms, more with bricks, and then even more with cut rock. Greek and Egyptian stone buildings are little more than stacking blocks, although they are very large blocks, often in very large stacks. These buildings were excellent examples of the biggest and best that could be built by the economies of the time; Greek temples pushed the boundaries of how tall pillars could be without good mortar, using primarily gypsum mortar, and the Egyptians went even further, building the biggest structures their economies could handle, with the size unlimited by materials or structural instability.1
Eventually, people started to realize that arches were more than just a fancy way of making doorways, and buildings started to use them as major structural elements. The Romans were perhaps the best at arches, and the Pantheon remains the world's largest unreinforced concrete dome ever built, and is still standing 2000 years later. However, during the
Renaissance arches and domes were rediscovered in a big way, and rich people who wanted to show off pushed them to the limits of the technology of the time.
Arches have big downsides. They have enormous upsides as well: they are very stable, can carry a lot of weight, and -- if you understand the rules -- scale very well. But there are some things that they can’t really do. If you build a nave 45 meters tall, you have a very nice cathedral, but you absolutely do not have a five story apartment building; dividing the arched space into separate floors without putting outward -- and therefore destructive -- force on the arch is a messy architectural puzzle. Knowing how to solve that puzzle will come sometime after a society learns a better way to make a five story apartment complex. So, what do you do with a one-room building that can be as tall as you like?
You brag. And here’s where politics and anthropology come into the equation. Kings and emperors can and do brag, and they do indeed commission arches and domes. But by definition, building the biggest thing takes the most time. The Great Pyramid of Giza took 20 years to construct; while bigger pyramids were possible, they would require the pharaohs to share pyramids across generations. Palaces have the same problem; kings don’t want to spend lots of money on something nice for their grandkids, they want the fancy digs now. God is much more patient. So that’s why cathedrals; because we can, and because a religion2 is patient enough to build something that takes more than a few decades. Without arches, cathedrals would be shaped like the Parthenon; without gods, cathedrals would be whatever could be built in 10-20 years, e.g., the Pantheon.
1. The pyramids, in Egypt and elsewhere, were limited by time and cost, but not much else; a pyramid made of limestone, given good foundations and smooth blocks, can grow as big as any mountain; no human has yet built a structure that can’t be supported on an appropriately-sized foundation of limestone blocks.
2. In practice, cathedrals were not built by monks, but by cities; religion just gave the cities the most socially acceptable way to brag. Or, if you prefer, city-states were the unit of religious congregation that were best suited to build cathedrals.