You have to go UP somewhere first, for example a tree or a roof, to fall off from there.
When you fall over, you are walking on the ground, but trip or slip on something.
I’d told Jack not to climb the tree. He fell off and broke his arm.
I was walking fast and the floor was wet, so I fell over and hurt my bottom.
You can also say: I fell over a box. = the box was on the floor and I tripped on it
