1. He's comparatively inexperienced. Paul is a first-term senator who has run and won only one political race in his life. Wait, you say, wasn't Obama also just a first-term senator when he ran for the presidency? Yes. But it wasn't his first campaign; he had run four times before making his presidential bid -- three successful state-senate races and one failed congressional bid. More important is the question of how Paul would do as a debater, as the rare non-lawyer in what is usually a field of legal sharpies. Clinton is an extremely accomplished debater; Paul might be able to become one over the course of a GOP primary contest, but it remains an open question how he'd fare.
2. He has a geographical handicap. There is no precedent for a Republican U.S. Senator from Paul's part of the South winning the presidency. In the 19th century, the GOP's presidential winners hailed from the North: Illinois, Ohio, New York. Midwestern and Northeasterners continued to dominate until the 1960s, when the GOP's base of presidential-level strength shifted to the West and the Southwestern part of the South, which is to say, California and Texas. Nixon was a U.S. senator and congressman from California before becoming vice president and president. Former California governor Ronald Reagan was born in the Midwest but presented as all Western, in his cowboy hat and boots, by the time he sought the presidency. Bush père was a congressman from Texas by way of a Northeast upbringing, later lifted into the highest office by the vice presidency. Bush fils was governor of Texas, but he had a good measure of Northeast culture in him, too. Paul was born in Pennsylvania and raised in Texas, but his two decades in Kentucky now situate him as a politician from one of the South's smaller states. Is this a moment where Kentucky, population 4.4 million, represents the future of America, the way California did under Reagan? No, it is not.
3. He's a sitting senator. Barack Obama won the presidency despite being a sitting senator, the first time that's happened since John F. Kennedy. How likely is it that the U.S. Senate, in this most divisive of eras, should now become the launchpad for presidential aspirations across the aisle, and that this less common pathway to the White House would be the one that works in the very next election cycle?
One could ask this question of Marco Rubio and Ted Cruz as well -- and also point out that it's been since Reagan that someone not currently in office, as Hillary Clinton currently is not, won the presidency.
4. Paul is an oppo researcher's dream. Having burst onto the national scene as a 2010 Tea Party politician, Paul might be able to win the Iowa Straw Poll in Ames -- assuming the Iowa GOP even keeps the straw poll -- but it means he also got his political career going with the aid of a fair number of folks who will be seen as extremists by middle-of-the-road political observers. The political conversation in Kentucky is very different from the one at the national level -- and so is the level of media scrutiny, as the Jack Hunter story shows. Paul has never been subjected to a national Democratic Party vetting and presidential-level media scrutiny. That's starting to change, and to reveal things that are going to make his political life more complicated.