They can mean the same thing, but they can also be different.
A "fuzzball" usually has a lot of fur or hair of some sort. A Persian kitten, who is chubby and round and covered with fur might be called a "fuzzball."
A "puffball," on the other hand, doesn't necessarily have to have fur or hair. It could just indicate something that is puffy and round (usually something that is not normally puffed up and round.) For example, when I had my wisdom teeth removed, my face swelled until my head looked almost completely round, like a basketball. I would've described myself as a "puffball" at that time.
Similarly, Dylan in your example was supposedly allergic to cats, so when he came into contact with cat fur, his eyes would redden and get teary, his nose would become congested and he would sneeze, and his cheeks would swell up (because his sinuses would be inflamed), making him look like a puffball.