Okay, I literally just signed up on this website, so I hope I am using this correctly. I would say to use the expression "run down by a car" would be used more when talking about a car specifically going after a person/thing/animal and ultimately running over them. For instance, if I were to walk outside and find my basketball smashed in the street, I would say "my basketball was run over by a car." If I watched a man drive his car intentionally toward a squirrel who was running across the street, I would say "that squirrel was run down by that car." For what it's worth, I am a US speaker, not UK. Also, just as a side note, in the US at least, it is "expressions" not "expressiones." Hope this helps! First post!

I'm a British English speaker, and for me they have the same meaning. My dog was run over by a car. My dog was run down by a car. I can't see any significant difference.