One difference is that you "cry" because of an emotional or physical stimulus. Your loved one dies (emotional stimulus), or you hit your thumb with a hammer (physical stimulus), and you cry. But you "weep" because of an emotional cause only.

As said by neoflight, it's a question of degree, but I wouldn't say "crying" is hysterical. I'd say that "crying" is more of a middle stage; a bridge between shedding tears and weeping. Another good word for "to cry" is "to bawl." It's like crying, but it's noisier.

It's a question of degree. Shed tears is perhaps only a few tears, crying is with lots of tears, and an almost hysterical state.

"Shed tears" and "Cry" are interchangeable, but "weep" sounds more archaic.

'Cry' is the most commonly used term. 'Shed tears' and 'weep' both sound more literary. You'd read them in literature or maybe in magazine articles, but we don't tend to use them in everyday speech.