As far as I'm concerned, they mean much the same thing.
Yesterday, I fell down.
Yesterday, I fell over.
However, I wouldn't say "I fell down" unless I was referring to a sudden collapse when perhaps I fainted or my legs gave way. I would use "I fell over" if I stumbled over my own feet, or tripped over something which was in my way.
The result of both is exactly the same. I started off in an upright position and I ended up on the floor.