What are the advantages of a more powerful console over the base model?
My understanding is that PS/Xbox games are designed so that they should work perfectly on either model
This assumption is wrong. They're not designed to work perfectly on any model; they're designed to work perfectly on one model and on the rest they merely work. The later models have more processing power; a game with graphics that were optimized for a later model will probably have a crappy frame rate on an earlier one.
It's also largely thanks to advertising: if customers think that the new model is the "hot new console", they're much more likely to buy it.