In DE, they give all of your capturable buildings more capture points. They also heal nearby units and you can train Elite Healers there of their specific culture. From the Greek ones you can train the healer hero Hippocrates (only once) and research Oracle of Delphi. I'd like to come up with healer heroes and 1 special tech for each cultural shrine.
In base game, they just heal nearby troops and you can train healers from them. They're basically capturable temples in that case, but with no techs.
It would be nice to be able to include these and other things, like Cultural Artifacts and Mercenary Camps, in random maps, toggle-able by the game host like how Catafalques can be turned off and on.
In DE, Cultural Artifacts give you a trickle of glory resource. In base game, perhaps a buff to capturing.
In DE, Mercenary Camps give you the ability to train 30 mercenaries, which cost 0 population, specific to the culture of the camp. There's also a camp which trains mercenaries specific to the civ you are playing. Each player has one of those civ-specific camps nearby, while the cultural camps, based on where the map represents, are scattered across the map.