Fear (Su)
A barbed devil's fear attack affects any creature it damages with its claws. A DC 20 Will save resists this effect, otherwise the victim becomes frightened for 1d4 rounds. This is a mind-affecting fear effect. The save DC is Charisma-based.
QUESTION:
If a PC makes this save the first time the devil damages them, do they need to make another save the next time the devil damages the PC?
Many fear effects state exactly what you did, that once a victim makes a saving throw he is immune to that effect, usually for 24 hours. This ability has no such wording. Neither does the Universal Monster Rule for "Fear".
Ergo, no immunity.
By way of comparison, if it had a poison effect, or paralysis, or flame, or acid, etc., you would want to apply those every time it attacks. Since this is a fear effect on a melee attack, it works like all those other kinds of extra damage/effect on melee attacks - for the sake of consistency and game balance.