They're used in different contexts. A spiderweb (or spider's web in GB English) is what the spider weaves. You're interested in the spider, and the fact that it has made the web fairly recently. If you meet cobweb, the spider has probably long gone. It's a question of poor housekeeping rather than arachnid behaviour. You're not interested in the spider, but it the fact that the room hasn't been cleaned for a long time. You'd complain to the hotel manager if your room was full of cobwebs. We also have idiom 'to blow away the cobwebs'.

They mean the same thing and people use them interchangeably.

They mean pretty much the same thing. A cobweb is the "tangled web" of the house spider, so it is a KIND of spiderweb. When I hear "spider web" I probably think first of the "orb web," the flat net-like webs with the spiral in the center, like the kind Charlotte weaves in "Charlotte's Web."