'weep' is related to reasons of grief and sadness, it is more intense than 'cry'. 'weep' is used more to describe a state rather than a momentary burst of tears ,where it is more appropriate to use 'cry' then. However there is another meaning of 'weep' You could say ,that a wound weeps, and it means liquid or blood comes from it because it is not healing properly. You can't use 'cry' in this case of course. 'Cry' also could mean 'to shout' and differs here totally in meaning .

You are on the right track. These words mean the same, however "weep" is more formal than "cry" and is associated with extreme emotion. " Weep" is rarely used in everyday spoken language. One would say "cry" more often. Examples: I would weep if someone close to me died. (Mourn;grieve;sob) I would cry if I broke a bone in my body. (Expressing physical pain or mild emotions such as happiness even) I will probably cry at my graduation. A baby will cry when he/she is hungry. She cried tears of joy at her wedding. I will weep when my mother dies. The use of "cry" is almost always correct, understood and accepted.The use of "weep" tends to sound archaic in most conversations.