The most straightforward indication of this is Paul Ryan's Catholicism. Atheism is a component of Objectivism and, consequently, a Catholic is decidedly not an Objectivist, as Ryan himself has noted.
Even humoring the premise that one can have major differences with the fundamentals of Objectivist philosophy but still share its political philosophy, Ryan has definite differences here as well. Objectivism calls for laissez-faire capitalism - that is, a complete separation of state and economics - and a government limited to the courts, police, and military. Ryan, to list merely the most prominent examples, voted for major government spending programs including TARP, the auto bailouts, and George W. Bush's expansion of Medicare. Yaron Brook, president of the Ayn Rand Institute, stated that he doesn't think “Ryan, even in his political philosophy, is as radical as I am or as radical as Ayn Rand would be.” Ryan is also – unsurpringly - “pro-life,” taking the opposite position of Ayn Rand on that issue.
While Ryan has stated that he voted for the aforementioned legislation because the alternative would be worse or because he wanted to be a team player for his party, it is doubtful that Objectivism, which so values principle and integrity, could call for such a strategy. Further, the policies he has actively advocated, including those in his budget proposals, are also decidedly un-Objectivist. These budgets – in content and by his stated intention – have been designed not to phase out, but to attempt to “save,” entitlements.
This does not mean Objectivism had no influence to Ryan or that Objectivists are hostile to him. Ryan has praised Ayn Rand and her influence on him, and prominent Objectivist publications have endorsed the Romney-Ryan ticket as non-ideal but worth voting, or even campaigning, for. The ideas of Ayn Rand have been a factor in the development of Paul Ryan's beliefs, and advocates of Ayn Rand's ideas appear pleased to see Paul Ryan on a major party ticket, but like the typical political influenced by a particular philosophy, Paul Ryan is no paragon of Objectivism.