I haven't read this book yet, and don't pretend to be up to speed with the philosophy and wider debates about animal rights, but I am aware that some philosophers and others identify at least one difference between children and animals which they regard as important: children are future adult humans whereas animals are not. Their capabilities and flourishing follow distinct paths. 
There is a good case for regarding non-human animals as different in significant ways from humans (but not necessarily of any less worth for that). There may also be a case for seeing children in different ways from how we often do in most 'modern' societies.
According to Hugh Brody's account, the Inuit believe their infants to be reincarnations of recently deceased grandparents. A mother may address her daughter as both 'daughter' and 'mother'. It would be impossible and silly to try and introduce such a belief into the industrial world. But we might have something to learn from the sense of trust, respect and reverence that such a belief brings with it for even the smallest and most vulnerable. Birth, life and death are greater than the individual ego.
Should only be sentient beings be worthy of 'special moral status'?
 See, for example Martha Nussbaum. A more radical view, perhaps, is taken by James Rachels.