The Dreaming was a monthly title inspired by Neil Gaiman's Sandman that ran for 60 issues (with one special, "Trial and Error") from June 1996 to May 2001.
The comic focused on the character of Lucifer (picking up after the end of the "Season of Mists" storyline) and had started as a Sandman Presents mini-series.
Sandman Mystery Theatre was a monthly title written by Matt Wagner and Steven T. Seagle featuring the Golden-Age Sandman Wesley Dodds in a film-noir-esque setting published by Vertigo for 70 issues between 1993 and 1999 as a direct result of Neil Gaiman's landmark series.
Initially The Dreaming, the first ongoing spin-off of The Sandman, served as anthology series telling mostly unconnected tales of various Sandman-related characters.
When it was decided that The Dreaming would be retooled into a more focused series with an ongoing storyline, it was decided that another outlet would be created to continue to tell anthology style stories.