Great question, Leo. Not quite. They can both mean in the backwards direction, but here’s the difference:
For the phrasal verb ‘back up’ (separable) we use it for several meanings:
1) To move in a backward direction.
He backed up and hit a lamp post
Careful when backing the car up.
2) To make a second (security) copy.
It’s important to periodically back up your computer files.
Today my computer crashed, but fortunately I backed it up yesterday.
3) To give support someone
The Senator backed up his fellow senator in the campaign.
When his boss accused her of wasting time, her colleague backed her up.
For the phrasal verb ‘back down’ (separable) we use it for two primary meanings:
1) To move in a backward direction, down in elevation.
He had to back the car down the hillin order to turn around.
While backing down the driveway he ran over his son’s bicycle.
2) To withdraw from a commitment or position. (inseparable)
When he was pressured to show evidence he backed down.
When the dog saw the man swinging the stick it backed down.