What's the difference between these two sentences?

We were practicing some basic sentence patterns in class today, one sentence we had to say was 私は日本語が上手じゃありません。I asked the sensei if "私の日本語が上手じゃありません" was the same and she said no, and answered in a way as to say that sentence made no sence. After class I asked her to clarify. She started saying the two sentences over in her mind to clarify which has lead me to believe there's some nuance difference between the two (rather than one being totally incorrect). She gave me an example saying that if I was speaking to a Japanese person and the Japanese person couldn't understand me then it was because 私の日本語が上手じゃありません, whereas 私は日本語が上手じゃありません has a different feeling to it. She had to explain all this in Japanese so I couldn't really get it, in the end she said not to worry at this stage, that I'll learn it all later. But I'm still curious, what is the difference between those sentences?

This doesn't really answer your question, but although I've personally never heard 私の日本語が上手じゃありません I think 私の日本語は上手じゃありません is perfectly acceptable. Maybe using が presents more of an implicit comparison between your Japanese and that of a native speaker (No, it's my Japanese that isn't good....

Actually, when I think back, maybe when I clarified it the second time I also asked if 私の日本語は上手じゃありません makes any sense also (which prompted her to start saying it to herself out loud)?? This would make sense because of the outright "iie" that I got when I first asked 私の日本語が上手じゃありません. So what would be the difference between 私の日本語は上手じゃありません and 私は日本語が上手じゃありません?

私の日本語は上手じゃありません My Japanese is not good.

and 私は日本語が上手じゃありません? I am not good at Japanese.

These are literal translations, of course in English the second is closer to "I don't have a talent for it" emphasizing the person that isn't skillful while "My XXX isn't good" holds more promise of improvement with practice. I sort of doubt the same distinction holds in Japanese, maybe 私は日本語が上手じゃありません is seen as more modest or polite ?

どちらでも使えます。どちらも使います。

また分からなかったら遠慮しないで先生に聞いてくださいね。

It seems that the difference is the sentence's subject. your first sentence puts more emphasis on "日本語" (私の only acts as a description for the subject, "my Japanese") while the other sentence makes you the subject. Just a subtle difference, but you should stick with the second one as it's more commonly used.