I don't think I'm aware of any substantial difference (in the 'vehicular' sense*), although there can be a difference in context. Here's an example, describing the same event:
Without a thought for the children, he just ran them down.
('ran them down' here suggests callousness on the part of the driver)
They were playing outside their house, and were run over.
(no value judgement about the driver)
(That is, the event described by the two phrases is the same, but the implications are different.)
I think there may also be a Br E/Am E distinction, but not a clear one. My impression is that 'over' is more common in Br E (that is, the difference of implication was introduced by borrowing a 'foreign' usage). For example, I use 'run down' more often than my grandfather did, but less often than Ouisch does.
* There are other meanings, where the two are totally distinct: Examples -
'Before the meeting, could I just run over the figures with you? I want to know what I'm talking about.'
'The shops are running down their stocks in preparation for the new winter styles.'
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