Except if you look at actual history of science and religion, it's way far from being that simple. Religions change, sometimes drastically, and are completely capable of overthrowing and even completely abolishing (see move from Catholicism to Protestantism) the authority structure. And science has plenty of authority issues, to the point where researches note that in some areas significant progress is made only when "old school" dies out and is replaced by "new school" - not because of reason, but because of physical limits of human existence. And some areas of science are very much influenced by political and other issues that have little to do with observation and reason.
So in practice both work in ways much more complicated than that.