If Mitt Romney is elected President of the United States, the president of the Mormon church will influence Mitt's list of potential cabinet members, court appointees, and his policies. Mitt's relationship with the president of his church cannot accurately be likened to John F. Kennedy's relationship to the Pope - not at all. Mitt is a bishop who presided over all of the bishops in the Boston area.
Of course, Mitt will do whatever President Thomas S. Monson, the living prophet on Earth, tells him to do. Such arrangements will all be hush-hush and under the table, but it will be the case.
Having lived in Utah for years, there are many good things I could say about Mormons and their culture. Nevertheless, I doubt Mitt Romney's ability to make good policy judgments because I think he lacks common sense. Mitt believes in celestial marriage, that in the next life he will be a god over his own universe with his god-wife beside him (exaltation). Mitt believes his heavenly father was once a mere man but was such a good man that he progressed to become God. Mitt's God is from the planet Kolob and has a God-wife, heavenly mother. Mitt believes Satan and Jesus are spirit brothers, spirit children of heavenly father and heavenly mother, as are you and I. And as a good temple-going Mormon, Mitt has been baptized in water in proxy for dead people in order for the deceased to have a chance at salvation (Baptism for the Dead). If you think I am misrepresenting Mitt Romney's beliefs and practices as a Mormon, you can easily educate yourself to Mormon beliefs in Mormonism for Dummies, written by Mormons and no doubt church-approved.
A presidential candidate's world-view and allegiances matter. Mormonism affects everything Mitt Romney does.