There are several differences between these two speeches. JFK was running in the general election, thus he was answering questions raised by the new possibility of having a Catholic president. Romney, though his choice of venue, the Bush library, allowed him to have presidential props around him, is not making this larger argument. Instead, he is making the argument that a Mormon can represent the Republican Party, a party very wedded to Evangelical Christianity. This fact influences the flavor of what follows.
JFK argues for a separation of church and state, he argues that politicians are obliged to pick judges and secretaries based on their fitness for office, regardless of their religious viewpoints; he further states in sum that the American electorate is best served when they do the same when picking a president. In short he states that presidents run countries, that is their function, and that being good at that is more important than their faith.
Romney focuses on American civil religion. He focuses on the lowest common denominator when it comes to faith. He is telling the Republican Party that its character is found not in the content of belief, but in the act of belief itself. The common movement of the spirit that goes on in any “spiritual person,” be they Muslim, Jew, Neo-Pagan, Christian, Hindu, Hari Krishna, or Mormon, is the bottom line that his Republican Party stands for.