Under the fear section of the glossary,
Frightened: Characters who are frightened are shaken, and in addition they flee from the source of their fear as quickly as they can. They can choose the paths of their flight. Other than that stipulation, once they are out of sight (or hearing) of the source of their fear, they can act as they want. If the duration of their fear continues, however, characters can be forced to flee if the source of their fear presents itself again. Characters unable to flee can fight (though they are still shaken).
Panicked: Characters who are panicked are shaken, and they run away from the source of their fear as quickly as they can, dropping whatever they are holding. Other than running away from the source, their paths are random. They flee from all other dangers that confront them rather than facing those dangers. Once they are out of sight (or hearing) of any source of danger, they can act as they want. Panicked characters cower if they are prevented from fleeing.
And the glossary definitions of frightened and panicked,
Frightened: A frightened creature flees from the source of its fear as best it can. If unable to flee, it may fight. A frightened creature takes a –2 penalty on all attack rolls, saving throws, skill checks, and ability checks. A frightened creature can use special abilities, including spells, to flee; indeed, the creature must use such means if they are the only way to escape. Frightened is like shaken, except that the creature must flee if possible. Panicked is a more extreme state of fear.
Panicked: A panicked creature must drop anything it holds and flee at top speed from the source of its fear, as well as any other dangers it encounters, along a random path. It can't take any other actions. In addition, the creature takes a –2 penalty on all saving throws, skill checks, and ability checks. If cornered, a panicked creature cowers and does not attack, typically using the total defense action in combat. A panicked creature can use special abilities, including spells, to flee; indeed, the creature must use such means if they are the only way to escape. Panicked is a more extreme state of fear than shaken or frightened.
So does "as quickly as they can" or "at top speed" indicate that they are requires to use the run action, or double move/withdraw if that would somehow put more distance between them and the source of their fear? Also, in the fear section definition of panic, it says the creature must "run." Does that mean the run action is mandatory even if teleporting/withdrawing/double moving would be "quicker?"
I've always interpreted these statements to mean: what will get the PC as far away from the source of the fear as far as possible in one round? Depending on circumstance, it might be a double move, the run action, dimension door, or even freaking plane shift :) Whatever it is, that's what the PC has to do (with the limitation that they won't jump to certain death off a bridge, for example).
That makes sense. To answer my own question, looking at the definitions again, it seems pretty clear that "run" doesn't mean the run action. If a panicked creature was required to use the run action, it couldn't use a spell/ability since it "can't take any other actions." But its clear that the creature is allowed to use spells/abilities.