When quoting, what's the difference between "someone says" and "as someone says"?

The distinction is in which fact is being asserted by the writer.

"X says that A" is a report by the author on what X says. The fact being presented is the fact that X said A, probably more than once. It's a description of habitual action by X.

"As X says, A" is the author saying A, very likely in agreement with X. The author is using X's habitual statement to backup the author's own stance on A, usually in an effort to bolster the credibility of that stance.

"Toni says" introduces a fact, that Toni has uttered the words that are to follow. "As Toni says" in addition indicates that the following words are germane to or supportive of a line of argument.