The world of White collar is purgatory, and Neal and Peter are incarnations of Kafka and Poe respectively
This is an afterlife in which people live out their deepest dreams. Neal and Peter represent all of the things that Kafka and Poe wish they were, but aren't.
Neal easily gets along with people, especially women, has tons of natural confidence, and is successful at what he does. In his fantasy, Mozzie represents his country doctor uncle, one of the few people he got along with, and Peter represents Max Brod, the only person who ever believed in him.
Peter has a long, stable relationship with his wife who has not died of a horrible disease, a position of power and influence, and is certain in his worldview.
And yet, some of the aspects of their original personalities come through, like Peter's mustache. Neal (or Kafka), in particular has difficulty fully believing in his own fantasy, which is why he can't form a lasting relationship, has trouble finishing anything, and is mostly estranged from his family. Only when they both work through the remainder of they're demons will they finally be able to move on.
