If a different character was to narrate the story, what aspects of the story would be different?

The narrator of Sebastian Faulks's novel Birdsong is not one person or character: Faulks employs a third person omniscient narrator, which means that the narrator knows everything but does not speak from the perspective of any one person. 
That being said, we should establish that the two main characters are Stephen, an English soldier fighting on the front lines during World War I, and his granddaughter, Elizabeth, who tries to piece together the details of her grandfather's history during a separate timeline in the 1970s.
Now, back to the question of the narrator. By its very definition, a third person omniscient narrator knows a lot about how the characters feel. However, this narrator is not actually experiencing the inner life of any of the characters: the narrator delves into their deeper thoughts, but he (she?) also makes outside observations and judgments about Stephen, Elizabeth, and the story's other characters. 
How would it be different if Stephen himself were the narrator? We would get a much richer, fuller idea of the traumas that he endures during the war. As it is, this book is tough to read: it is emotionally draining to read about Stephen losing his best friend on the battlefield, losing the love of his life, and so on. However, because the narrator is not actually inside Stephen's mind, we get an overview of these events that is more distant than it would be if Stephen were the narrator. There is a summary of the pain that Stephen experiences in the trenches, but there is much that we are left to imagine. 

No child or future generation will ever know what this was like.  They will never understand.  When it is over we will go quietly among the living, and we will not tell them.  We will seal what we have seen in the silence of our hearts and no words will reach us.

This passage is directed at us as readers. Because of Faulks's choice of narrator, we can guess what a soldier's traumas might have been like, but we will never truly know. We "will never understand."