Rand Paul is not splitting with his dad on this issue—the Paul family seems dead-set against contraception. Not just government funding of contraception, but contraception in general. Ron Paul says birth control doesn't create immorality, rather that "immorality creates the problem of wanting to use the pills. So you don’t blame the pills." In other words, he's not for banning contraception, but he is calling contraception "immoral." And he also believes that government shouldn't pay for it.
It's interesting to me, because Rand Paul is essentially taking the stance that a corporation's liberty is worth more than an individual's liberty. (You can argue that if someone doesn't like Hobby Lobby's policies, they're welcome to go find another job elsewhere, or to start their own hobby store, but the truth is that's a disingenuous argument—the market can't sustain an infinite number of hobby stores, and any given area only provides so many jobs. There are only so many options, and people don't need birth control at some random point in the future. They need birth control when they need it.) He's not so much a libertarian as he is a corporatist, which is an inelegant word to describe someone who argues on behalf of corporations. It's unfortunate that corporatist is a word that has basically opposite meanings, depending on who uses it. I would love to see someone come up with a new word to define political interests—like Mitt Romney, like the majority of the Supreme Court—who advocate for corporations as basically super-humans, with amplified powers of free speech and religious freedom that impinge on the rights and freedom of average Americans. At this point, it is most definitely a real political movement, and political movements need names.