What is the difference between predatory publishers and vanity presses? The University of Sunshine Coast's website gives the difference as:

Vanity publishers are publishers that will charge the author a fee for publishing a book ...
While similar to vanity publishing, predatory Open Access publishers seek to take advantage of the Gold Open Access model, whereby the author pays to have an article available as Open Access on the journal website.

This sounds pretty trivial: vanity presses are presumably willing to publish anything, including journal articles, if paid.
If there is a difference, what is it? If there is no difference, why did Jeffrey Beall create a new term?
I've always seen the terms as somewhat distinct. 

A predatory journal seeks to deceive and create the impression of publishing a peer-reviewed article. I take the predatory journals to always be engaged in a conspiracy to defraud the academic process (where authors can range from stooges to willing co-conspirators). Here, the reason why authors pay can be imitation of Gold-standard open-access or it can be pay-to-play to defraud their employers.

In contrast, vanity presses are sometimes just really low-tier publishers without malice. I don't think they always are trying to create the impression that what they produce is high quality academic work. The failing from an academic perspective of a vanity press is that they don't have robust (or any?) standards for what they will publish in book form and they don't engage in academically sufficient (or any?) editing practices of what they publish. The product on offer is you can say "I published a book". I think the business model here is that vanity publications don't make publishers money, so they instead charge the authors to break even (or make a profit?).

Regarding your point about a vanity presses willingness to publish an article, sure they can print it up for you -- as a book, but I severely doubt they are going to make it seem like it's volume 4 of the international journal of basketweaving and wickerwork.

I'm sure it varies by field, but in my field as I understand it, journal articles in top-tier journals tell us far more about the quality of someone's work than even books in top-tier presses, because presses publish things of sufficient quality that make them money. As you go down the scale on each side, incentives and motives shift, but the article vs. book distinction remains meaningful.