There's not a difference by themselves. In both cases, someone other than you cuts your hair. However, when you add a person to the sentence, the grammar does change: I have my mother cut my hair. I get my mother to cut my hair.

I think (I have my hair cut.) means that I cut it myself I get my hair cut means that it is cut by a barber or someone else.I am not sure whether it is true or not :)