Data publishing is the publishing of a dataset not tied to an article. The current practice is to deposit your data in a repository and link it to an published article in a journal. This method is to publish your data as a “data paper” which describes a dataset in detail, without analysis or discussion. They can also be linked to published articles in traditional journals.
Data journals are publications whose primary purpose is to expose datasets. They enable the author to focus on the data itself, rather than producing an extensive analysis of the data which occurs in the traditional journal model.
They are published in the “Data Papers” section of an established journal, or in a journal dedicated to data papers:
PANGEA: Data Publisher: They manage an assortment of platforms and journals.
ANDS has a very good page about Data and Journals.