About/over in given context

After we broke up, she cried about me to her friends.

After we broke up, she cried over me to her friends.

Is using about grammatically incorrect here? Is there a difference in meaning between these sentences?

They are both correct. However there is a niche difference you ought to know.

If you cry about me to your friends, typically, you are talking about our relationship while crying.

If you cry over me to your friends, typically, you are just sobbing.

Even though there are interchangeable, you might want to use about in some situations and over in others.