The difference between the religions now is that they are hooked to a specific technology. To see why this is a difference, instead of thinking of them as the real world religions of Christianity, Islam, etc. think of them this way:
Religion You Can Get When You Get Philosophy
Religion You Can Get When You Get Theology
Religion You Can Get When you Get Divine Right
etc.
Or you could just rename them like this:
Philosophy Religion
Theology Religion
Divine Right Religion
Meditation Religion
etc.
This makes the religions, strategically distinct even if there's no distinction in terms of religion-based doctrine, etc.
To get rid of this small small diversity would make it 100% bland as opposed to 96% bland.
You can implement the stuff you're talking about even with Civ 3 ... so that's a separate thing.