It matters because their convictions will ultimately and inevitably govern their decisions and actions. The beliefs we hold most dear cannot help but influence our thought processes. Given the awesome power of the US presidency, the central task of which is to make decisions that influence the entire country, and even the entire world, do we not want someone in the office who is adept at considering evidence, exercising reasoning, and arriving at sound choices?
end tyranny in Iraq'. When a president's religious beliefs commit the country to military intervention overseas, they couldn't possibly be more relevant. When a president's religious convictions lead them to reject scientific evidence in favor of ancient scriptural doctrine (witness the Republican candidates in the video professing disbelief in the theory of evolution), they couldn't be more relevant.
There are other implications as well that are truly frightening. If a president believes that the Bible is inerrant when it comes to Genesis, then are they equally convinced of the veracity of Revelations? Given the authority of the president over the world's most powerful military--including the world's deadliest nuclear arsenal--the degree to which he (or she) embraces apocalyptic prophecy becomes exceedingly relevant, does it not?
If Mitt Romney truly believes that he has magic underwear and that he will be inheriting his own planet upon his death (as in his Mormon doctrine), or that the Milky Way was ruled 14 trillion years ago by an intergalactic warlord named Xenu (Scientology), or that two of every animal could be housed for forty days and nights on a ship smaller than the Titanic (Christianity), I don't see how anyone can, with a straight face, profess that this would not call into question their reasoning in other areas. Has anyone ever known someone who believed in alien abductions, or that the moon landing was a hoax, or that Elvis was still alive, who was stone cold rational in every other area of life?