Most Fear/Frighten effects in the game confer additional effects beyond what the condition itself says
Charm and Fear are both similar in this regard in that the 5th edition rules have a multitude of spells, class features, abilities, etc., that can imbue the Frightened condition on a character; and almost all of these features have specific stipulations about how the Frightened condition affects the target above and beyond what the condition itself provides.
- Cause Fear (XGE, pg. 151): Causes the Frightened condition but doesn't say anything else should happen.
- Eyebite (PHB, pg. 238): One of the effects confers the Frightened condition; the target must use their action each turn to Dash away from you as fast as possible, but the effect ends if they reach 60 ft. away or more.
- Phantasmal Killer (PHB, pg. 265): The target takes additional damage at the end of each of their turns while the Frightened condition persists.
- Symbol (PHB, pg. 280): If the fear effect is used, the target is Frightened, must drop items they're carrying, and must move at least 30' away from the origin if able.
All of these use the Frightened condition, all of these have the standard effects of being Frightened (Disadvantage on checks, unable to move nearer to source) along with whatever effects the specific spell/effect has, and all of these effects end when their respective instance of the condition ends.
So when you see this kind of additional effect, like what you see with Abjure Enemy, it should be interpreted as causing that effect in addition to the Frightened condition, not instead of it or its effects.