The most immediate similarity between both works is that they offer a glimpse of the Holocaust through the eyes of young people. Anne is an adolescent and Bruno is a young boy. Both of them view the world with authenticity and openness. They follow their own code of moral right and wrong and use this to examine the world around them. At the same time, it is through this lens of perception that both are able to make critical judgments about the world that envelops them. Anne is able to raise questions and engage in reflection about a world in which something like the Holocaust would happen. Bruno's questions are critical in their nature. He might not intend for them to critically assess the world around him, but his inability to pronounce words and ask questions for which easy answers are not evident end up accomplishing this goal.
I would say that a fundamental difference between both works is the characterizations of each young person. Anne is an active participant in understanding the world around her. Yet, her role for action is limited. Her action is constrained to a condition of thought and a sense of seeking to comprehend the world around her. She is not able to really do much in her diary that exists outside of her own thoughts and questions about a world gone mad. Bruno is able to take direct action in his aiding of Shmuel. Bruno has an opportunity to undertake "a great adventure." While there is fear, he overcomes this and actually supports his friend until the very end. While Boyne's work is historical fiction and Frank's diary is nonfiction, this would represent a significant difference in their characterizations.