Yazoo and Yaz are actually the same band. They had to use the name Yaz in the U.S. because of copyright issues.
Vincent Clarke of Yazoo was actually one of the founding members of Depeche Mode, but left after their first album and enlisted Moyet to create Yazoo.
After two albums, Clarke and Moyet split. Alison Moyet went on to pursue a solo career. Vince Clarke continued to create other projects/bands, most notably Erasure.

Yazoo - Situation - 7 Inch Yaz Depeche Mode on Mute Records