Probably quite a lot. Mormonism has some peculiar attributes about it, one of them being that thing about how people will become de facto “gods” in the hereafter. It has a certain, but very odd, Calvinistic flavor to it. If one becomes important on earth, has lots of progeny, gives a lot to the Mormon church, one might rule a whole galaxy as its god hereafter. If, however, one is just an ordinary Mormon who gives his service and 10% of a smallish income to the church, he might end up with just one modest planet to rule someday.
And so, it has an element of “success=election” aspect to it. Romney has probably given 10% of everything he ever made to the Mormon church, and that would be a lot. If so, he probably has a high station in it, officially or unofficially, and his word probably means a lot more than we would think. It would be incorrect to compare Mormonism with Shinto, but there is one thing they share to a degree. The higher a Mormon is in the church, the more godlike become his pronouncements. So, for example, the pronouncements of the President of the church become doctrine, and following those precepts is spiritually mandatory in belief and in action.
So, while Romney is not the President of the church, if he’s one of the really high-up guys, his word isn’t law, but it has some aspects of spiritual mandate to it.
That’s not to say Cruz has no other appeal to Mormons. But Romney’s backing should not be underemphasized in its importance.