Why Did Tobi Join the Akatsuki

Why did Obito join the Akatsuki?

He was forced to join this organization as he killed his entire clan for the sake of peace and then fled the village. The best place for him to go to was the Akatsuki, even more so because they were a growing threat and Itachi could keep them in check from the inside.

Why did Tobi let Itachi join Akatsuki?

He let him stay because a lot members of the organization had their own motives and knew they would betray the overall effort at some point or another. They are all missing nins which means loyalty to an organization was clearly not on the top of their list. Many were even there just to spy or further their own goals.