What's the difference between a naturalized U.S. citizen and one born outside the U.S. to an American parent?

U.S. law defines a naturalized citizen as one who became a U.S. citizen after birth. It be through many ways: by applying for naturalization, or by derivative citizenship from a parent while in the U.S. A person born abroad to a U.S. citizen parent is in most, but not all, cases, a U.S. citizen at birth. If they are a U.S. citizen at birth, then by definition they are not a naturalized citizen. Therefore, the two are very different and mostly mutually exclusive. In the rare case they are not a U.S. citizen at birth, and later becomes a citizen, then they are also a naturalized citizen. I will assume you are talking about people who are born abroad, and who are U.S. citizens at birth. Interestingly, the two categories of people have different constitutional advantages over each other. An advantage that people born abroad who are citizens at birth have over naturalized citizens is the ability to be president. The U.S. Constitution requires the President to be a "natural-born citizen", which most scholars believe means someone who has been a U.S. citizen since birth. Therefore, this means all U.S. citizens who are not naturalized citizens. On the other hand, an advantage naturalized citizens over people born abroad who are citizens at birth, is the protection from losing their citizenship involuntarily. The 14th amendment provides citizenship to people "born or naturalized in the United States, and subject to the jurisdiction thereof", which the Supreme Court has ruled in Afroyim v. Rusk means these people's citizenship cannot be taken away involuntarily by law. However, people born abroad who are citizens at birth, are neither born in the U.S., nor naturalized in the U.S., and therefore their citizenship is not constitutionally protected. The Supreme Court ruled in Rogers v. Bellei that a law can take away citizenship involuntarily from someone born abroad, who was a citizen from birth.

That is the only place it's different, however, there's a chance that security clearance might be affected...actually, I doubt that, there have been plenty (or at least some) cabinet secretaries who were foreign-born, naturalized citizens.

The one born outside the US to an American parent can become president, the naturalized one can't.

A person born outside the United States to an American parent usually automatically acquires U.S. Citizenship at birth under 8 USC 1401.  This is a NATURALIZATION statute and one who gains citizenship under a naturalization statute is by definition, a naturalized citizen.  While it is popularly believed that naturalization is a process that takes place sometime after birth, that is not true.  Here's an opinions offered on this subject in dicta by a Supreme Court Justice: "Although those Americans who acquire their citizenship under statutes conferring citizenship on the foreign-born children of citizens are not popularly thought of as naturalized citizens, the use of the word "naturalize" in this way has a considerable constitutional history. Congress is empowered by the Constitution to "establish an uniform Rule of Naturalization," Art. I, § 8. Anyone acquiring citizenship solely under the exercise of this power is, constitutionally speaking, a naturalized citizen. -Rogers v. Bellei, 401 U.S. 815 (1971)  

The difference is that the one born abroad to the American parent becomes a US at birth  - that is by the operation of law, but not always. What that happens, the person's birth is reported to the US consulate and a Certificate/Declaration of Birth Abroad is issued by the US government. Such US citizen is considered a natural born US citizen who can seek any office in the USA including that of the President. Naturalized citizens are not considered natural born citizens and are not qualified to seek US presidency. They can seek any other office in the land except the Presidency and by extension the Vice Presidency.

The suggestion "security clearances"  was correctly doubted.  The issue there is continuing close contact with foreign nationals.