Cobwebs and spider webs are inherently the same except that cobwebs are older. A spider web is a form of webbing that is created by a spider and spun from spider silk. Its primary purpose is to trap their prey for food. It becomes a place in which the spider finds a home.
Webs that are currently being used by the spider are called spider webs. When the spider decides it is time to relocate or it dies, the result is a cobweb, especially if it has been years since the spider spun its web. Sometimes a spider web has existed for a long time and has gone through several transformations, and it becomes a cobweb.