In a sense our separation began with the emergence of life. A set of complex molecules divided themselves off inside a membrane and began to reproduce in defiance of the second law of thermodynamics. This states that entropy or disorder must always increase. Life defied this law, but she could only do so by exporting disorder to the surroundings from which she drew energy. The development of nervous systems was the second step.
Higher animals have self-centred impulses, to get food, to avoid danger, to reproduce.
They don't see themselves as part of nature or of a species with a common interest. Apart from their close siblings nature seems to them a huge bag of promises, threats and competitors in their own drive to survive and multiply. To a lion, hunger makes the whole horizon look like dinner. To an antelope, fear makes every bush seem like a predator. Humans have the most fully developed intelligence and self-awareness of all animals.