Rift sounds like rip, and it's helpful to think of it that way. When there is a rift in a political party over a particular policy, it is like a rip or tear in the fabric of the group.
Rift has a slightly different sense than break. A group can have a rift without having fully broken apart. A rift can heal, though a break is permanent. A rift can arise between sisters over whose turn it is to do dishes; it can quickly heal when parents promise an ice cream trip when the dishes are done. The Earth's crust has rifts, where it's splitting apart, like in East Africa.