The first sentence is wrong and the second is right - that's the difference. 'Why were you broken with your girlfriend?' doesn't mean anything. It isn't proper English - forget about about it. The second sentence is fine.

"Why did you break up with your girlfriend?" is the proper phrase. "Why were you broken with your girlfriend?" is not correct or clear. The meaning isn't about separating, but closer to "emotionally damaged" or "spiritually conquered". It's an awkward sentence, so please don't use it.

The first is a bit broken in fluency, but second sounds better. second is better understood but I think th. Meaning is same for both.