The meaning of "cry" as opposed to "weep" are about the same. However, I think "cry" refers more to the sound that is made and "weep" refers more to the streaming of tears down someone's face. You can also use "weep" and "cry" in other contexts. For example, a "weeping wound" would be a wound that excretes a small amount of fluid, such as pus or blood. As for "cry," it sometimes means to simply yell. For example, in a book you might see a sentence like: "'Get out!!' he cried." This means that he yelled out of anger, rather than cried from sadness. I hope this makes sense. Let me know if there is anything that needs clarification. Good luck!

Weep is not common in American English. I would use cry for regular crying and sob for severe crying.

Not much, at least to me. They can be used pretty interchangeably, though I would say, I hear 'cry' used more commonly where I live.