"She bit into an apple." This is grammatical and makes sense on its own. In the story of Snow White, biting into a poisoned apple is an important part of the story.
Snow White bit into the fruit, and as she did, fell to the ground in a faint: the effect of the terrible poison left her lifeless instantaneously. (Source)
"She bit off an apple." doesn't make sense without more context. Usually we would say something like "She bit off a piece of apple." To "bite off", a piece needs to be separated from the whole with your teeth. If there was an edible apple tree like the one below, she could bite off an apple (from that tree).
There is an idiom that might help explain it - to bite off more than you can chew. Taken literally, it means that the piece you bit off is so big, it fills your whole mouth making it hard to close your jaw to chew it. I picture something like this
Some more examples:
The dog bit into the man's leg. (Doesn't tell you if the dog has let go, or if the dog has taken a chunk out of the leg.)
The dog bit off the man's nose. (Most or all of the man's nose was separated from his face. Ouch!)
The dog bit the man. (Doesn't tell you where the dog bit the man, or whether it bit anything off, or how severe the bite was.)