They're more like cousin cults rather than rivals - one is esoteric (select few) and the other is exoteric (for the masses), yet they're mostly similar when you get down to it. Safe to say the Catholic Church absorbed what was left of the cult once it became the official state religion. Now there was no need for it in the military since you could just replace Mithras with Jesus (same dying-resurrecting god/bringer of the new age/similar rituals, etc.), sprinkle some Augustinian just war theory, and you're good to go, as if nothing changed. I view the relationship more like how the Mormons appropriated Freemason rituals into their religion, yet Smith was a part of the Anti-Masonic movement. Mormonism, like Christianity, is for the masses, and Freemasonry, a lot like Mithraism, is a mystery school cult.