The Saudis want to control all of Yemen. While Yemen is geographically smaller and dirt poor it has an equal number of citizens and some valuable resources. The Saudis have for decades financed Wahhabi preachers to proselyte in Yemen. But Yemen has its own milder style of Islam and the Wahhabis were generally not welcome. There are also plans for a Saudi pipeline to Yemeni ports which would allow Saudi exports to bypass the Strait of Hormuz. For their war on Yemen the Saudis allied with the United Arab Emirates.
The UAE have their own plans for Yemen. They wants to control south Yemen and the port of Aden as a gemstone in their expansionary Dubai Port World conglomerate.