Which is the correct form in the following two sentences:

"while the car was driven by..."

"while the car was being driven by..."

and if the two are correct, what is the difference between them?
Anonymouswhile the car was driven by..."
"while the car was being driven by...
They're both correct and natural.

Both clauses are past tense, passive voice.

The first is past simple. The second one is "past continuous."

The continuous tense is not really necessary, because context and the (semantic) meaning of the verb make it obvious that the action is continuous.

It does, however, tend to throw the focus more on the (unmentioned) actor/driver.

Casually, we sometimes say, "The car was driving by slowly." (Past continuous, active voice, intransitive)

How does the car do that?? Very little attention is directed toward the driver.

BTW, I hope you don't mean, "While the car was being driven byJack, Jill was asleep in the back seat." Emotion: thinking
BTW, I hope you don't mean, "While the car was being driven by Jack, Jill was asleep in the back seat."

Thank you for your detailed answer, but I've actually meant the above example that you provided.

Please advise if that changes anything...

Thank you
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While the car was being driven by Jack, Jill was asleep in the back seat.-- This is fine, too. I think Avangi was having a little joke.
AnonymousPlease advise if that changes anything...
It doesn't change anything. Both versions are fine with or without "being."

Note that "by" can have two different meanings:

The second car was driven by a woman.

Another car was driven by. (Another car was driven past.) (Somebody drove another car past.)
Aha! I just understood what you were saying, Avangi!

While the car was being driven by Jack, Jill was asleep in the back seat. -- the normal interpretation is that Jack was driving the car, but it can also be construed as the car was being driven past Jack (who is perhaps standing in the road).
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You give me too much credit, sir.

My intention was as you originally suspected.

But now that you mention it, we might have three senses here:

(1) to drive by (a fixed expression/idiom) The car was driven by on two separate occasions. (The car drove past.)

(2) to drive by something The car was driven by the school. (Somebody drove the car past the school.)

(3) to be driven by somebody The car was driven by Jack. (Jack was driving the car.)