What is the difference between these sentences including present participle?
The running car has no batteries.
The car (which is) running has no batteries.
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Your question is a bit unclear. Is the second sentence "The car running has not batteries." or "The car which is running has no batteries."? If it's the first one, it's grammatically incorrect; adjectives should precede nouns, so "car running" is incorrect, and should be corrected to "running car", as in your other version of sentence.
But if your question is about "The car which is running", then the difference is really just a stylistic one; "the running car" and "the car which is running" both mean the same thing, but some speakers or writers may prefer the sound of one over the other.
Personally, I think both phrasings are actually a bit awkward, because "running car" is not a phrasing I hear very often; if I needed to refer to a running car, depending on context, I might actually say "the car that is running", especially if I'm picking one car out of a row of several cars, e.g. "There are three cars out front; the car that is running has no batteries."