The basic meaning of verarbeiten is to use raw materials to produce something more elaborate, where the focus lies on the transformation of substances to a new substance:
Sie verarbeitete die Zutaten zu einem Kuchenteig.
This meaning has also been applied to psychological processes of coping with traumatic experiences, as they, too, imply a development and the aspect of change for the better.
Aufarbeiten, on the other hand, basically means working or finishing off a task and the material upon which the task is bearing. So, the focus lies on getting something out of the way:
Er arbeitete alle Akten auf, die sich auf seinem Schreibtisch angesammelt hatten.
(This meaning of aufarbeiten is understood and common in the whole of the German language area, whereas another meaning, refurbishing or renovating something, is not; to my knowledge, you would use it mainly in the nothern half of Germany.)
As Kilian Foth already said, both your examples are well chosen. Whereas both verarbeiten and aufarbeiten are used for coming to terms with one's past, the former explicitly refers to a mental process which is active but not necessarily shows on the outside:
Eva versucht, ihre schreckliche Vergangenheit zu verarbeiten.
The general idea is to come to terms with a traumatic impact in order to neutralize or contain it one way or another so it can't any longer do damage to a persons mind and soul on a subconscious level, whereas it is not possible to clear the trauma itself.
Aufarbeiten rather signifies an action where the past in some way materializes you have or want to cope with, by writing a book about it, discussing it publicly, arranging and commenting memorablila, etc., so you can close the case afterwards without remorse:
Er überlegt, seine Kriegserlebnisse in einem Artikel aufzuarbeiten.
"Aufarbeiten" doesn't necessarily carry the implication of a psychological trauma, it may also just be a process of "sorting out" one's memory in order to get a clear picture of what previously had consisted of a mashup of memories without consistent chronology, not clearly related to facts and figures. Of course, this can always have a kind of cathartic impact as it is intended in a medical context. From this viewpoint, as Kilian Foth already pointed out, "aufarbeiten" and "verarbeiten" are two sides of the same coin.
This is why, listening to and reading everyday communication, you wouldn't always be able to pinpoint them to a special meaning as distinguished here, as they are often used generally as synonyms without further consideration.
