Why are we using off here? What additional meaning does it add?
The meaning of off from http://tfd.com/off that suits this sentence is: "away (from a place, time etc.)".
There's more than a hair's breadth of difference between these two statements!
The first simply means she had a haircut. Maybe she is now sporting a new style, or maybe she just had a trim.
The second means that her hair has been cut drastically short, if not completely shaved. It might be used if she once had very long hair, but, after her latest trip to the salon, it's now cut very short.
Note that the difference between cut and cut off can vary, depending on what's being cut.
She cut her finger means that she's bleeding, and may need to put a bandage on it. She cut her finger off means that, unless a doctor performs reattachment surgery, she'll only have four fingers for the rest of her life. He cut five pages from the back of the book might suggest an editor opted for a shorter ending. He cut off five pages from the back of the book could mean a bookbinder removed physical pages with an exacto knife.
For hair, however, she cut her hair is simply a common expression meaning she got a haircut. She cut her hair off suggests something far more drastic, which I've already explained.
In English, the phrasal verb cut off is not the same as the verb cut. Cut certainly means "shorten, shave, shear, or slice". But off is not a preposition in the phrasal verb cut off - instead, the off modifies the meaning of the main verb. In this case off means something like "completely, drastically".
For another example, look at eat up. When you eat a hamburger, you simply ingest all or part of it. When you eat up a hamburger, you completely devour it.
The up in eat up does not mean "above, higher, or toward the sky". Similarly, the off in cut off does not mean "away", it means "completely".
The general meaning of cut off is "remove (something) from something larger by using a sharp implement."