Full Faith and Credit is different from reciprocity. Reciprocity is a matter of agreement between states, on matters as to which the FF&C clause does not apply. FF&C is basically "automatic" reciprocity -- no need for an agreement.
The driver's license example is a matter of reciprocity, not FF&C -- the states have agreed to accept each others' license as a matter of convenience to citizens, notwithstanding differing state interests that could legitimately prevent FF&C from being applicable.
On matters of federal law at a minimum, FF&C requires states to honor other states' acts, records and judgments. So, if state A were to determine that a person has met the conditions necessary to exercise a federal right, state B must accept state A's decision as its own. FF&C is somewhat broader than that, but beyond federal rights it is easy to venture into the fuzzy realm of states not having the right to impose value judgments on others.
If the Supreme Court were to determine that there is a federal constitutional right for people of the same sex to marry, for example, then under FF&C any state would be obligated to honor a marriage from another state. Unless and until such a federal right is established, however, it is piecemeal at the state level, by state declarations of that right and reciprocity agreements between states.
This is similar to 2nd Amendment rights. Because the basic human right of armed self defense is a federal right, state A's determination that a person is qualified to exercise that federal right (the person is not a felon, not mentally ill, etc.) is binding on every other state. So, under FF&C a person with a concealed weapons "permit" should be able to exercise that federal right in any other state. But the states have not figured this out yet, and they are focusing on reciprocity agreements rather than FF&C.
I personally believe that these 2 issues establish a great litmus test for intellectual integrity. People on the left typically will say FF&C should be honored for same-sex marriage, while people on the right will typically say that FF&C should be honored for concealed carry. But it takes a particularly high level of integrity to admit that if one's "pet" issue should be honored under FF&C, then so should the other. Test your friends with this, it can be fun to observe the frustration.