Is there a difference between Pin<Box<T>> and Box<Pin<T>>?

Pin<Box<T>> is what you want. Box<Pin<T>> will not work at all.

Pin requires its type to be a pointer of some kind. It then prevents you from moving out of this pointer (if the pointee isn't Unpin), by requiring unsafe to access it mutably. In Pin<Box<T>> Box<T> is the pointer. It is common because you can create it safely (as opposed to Pin<&mut T> that without macros can only be created unsafely) because you give the ownership of the Box to it, and thus you cannot access the inner T not through the Pin. Box<Pin<T>>, on the other hand, is useless: it is impossible to create if T does not implement Deref (as Pin's constructors require that, because they are meant to use with pointers) and even if T does, the Box is redundant: you already have a pointer, there is no need to wrap it in Box. In addition, you cannot create an instance of Box<Pin<T>> if the <T as Deref>::Target does not implement Unpin without unsafe code, and there is little benefit in Pin with Unpin types (it can be passed to APIs that require it, such as Future::poll(), but in that case you don't need the Box).