In both cases, we would use the first sentence as a direct report, and the second sentence may be used as a clarification.
A: "Rupert says that dogs make a lot of mess on the beach."
B: "Does that mean that owners should clean up after their dogs?"
A: "No, Rupert is saying that dogs shouldn't be allowed on the beach at all."
We use this kind of construction a lot when referring to ourselves. For example, sentences like:
"What I would like to say is this: kittens are cute"
"I'm trying to say that I love you, but I'm too shy"
"I'm not saying that I hate pizza, just when it has too much cheese on it."
One main reason for the difference in meanings between 'says' and 'is saying' is that 'says' doesn't refer to a specific timepoint, while 'is saying' is specifically talking about right now. For example, even if I haven't talked to my father in a week, it is still valid to write:
"My father says that eating too much pizza makes you fat"
Because it is something that he has said in the past (that I believe is still true). Similarly, we might use 'says' to describe what has been written (in the past) in a book, even though the person clearly isn't saying it.