Other animals can be altruistic, but Gazzaniga argues that they fall far short of the human moral sense. He also explains how humans deal with far more complex social issues that other animals - you might think of social insects, but they tend to have rigid behaviour patterns. Humans also have a storng sense of empathy - if you seen someone getting hurt you are likely to experience some of their pain. Later chapters of the book look at the multi-level theory of mind which humans possess, and the final section looks at the possibilities of future enhancements of the human mind.
I felt, though, that the book wasn't structured in a way to convince the reader of Gazzaniga's arguments. The first chapter jumps into describing some of the latest work in neuroscience, and so ends up being over-technical. It gets easier later on, but tends to wander around - a couple of pages after out-of-the-body experiences we find out about the evolution of dogs, without really seeing what points are being made. On the other hand,the book does have lots of interesting material and extensive notes, so if you want an introduction to what makes us human then this might be of interest to you.