They are very close in meaning. I am a well-educated native speaker and this is the sort of question that sends me to the dictionary. My personal favorite is the American Heritage Dictionary online, because the definitions are very clear, and sometimes it has usage notes. It has one for "frightened."
"Synonyms: frighten, scare, alarm, terrify, terrorize, startle, panic
These verbs mean to cause a person to experience fear.
_Frighten_ and the more informal _scare_ are the most widely applicable: "The Count's mysterious warning frightened me at the time" (Bram Stoker). We scared each other telling ghost stories before bed.
_Alarm_ implies a state of fearful anxiety, often brought on suddenly: The sight of the approaching shark alarmed the swimmers.
_Terrify_ implies overwhelming, often paralyzing fear: "It is the coming of death that terrifies me" (Oscar Wilde).
_Startle_ suggests a momentary fright that may cause a sudden, involuntary movement of the body: The clap of thunder startled us.
_Panic_ implies sudden frantic fear that often impairs self-control and rationality: The realistic radio drama panicked the listeners who tuned in after it had begun."
It doesn't include "horrified." "Horror" adds the idea of disgust. For example, if you turned on the light in the kitchen and saw fifty cockroaches on the floor, you would be horrified. You wouldn't necessarily be frightened or scared, at least not very frightened, because you know that cockroaches don't bite. You might be startled, because you weren't expecting it.