It's a simple problem with a non-trivial solution.
How a browser works -- a browser downloads the .html file (like you are doing in cURL). It then *parses* that file into element nodes and linked resources (like images, javascript, etc.) as well as layout and all that jazz. It then requests each of these linked resources as individual files and uses them in the display presented to the client.
It is that (second to) last bit which is the step you are missing "It then requests each of these linked resources as individual files..." and I think you recognize this.
cURL doesn't have a built in parsing mechanism for the page that you request through it. You will need to pass that page's source to either a DOMDocument or XML parsing method and then grab all the types of linked resources you want from there. Or you can do a regex type search for "http, src, etc." strings in the page to pull the URLs out that way.
Cheers