To 'cry' is to shed tears, it could be by any reasons whether physical pain, sadness or whatever,it is a more general term . To 'weep is to shed tears as well but as an expression of grief or unhappiness in particular.

Weep is a much more emotional word. It is generally not used in conversation.

They are completely interchangeable. The only difference is that weep implies felling sorrow and showing sadness, while crying is more associated with tears running down one's face. Cry also has a second meaning which is to yell.