Animals compete with and kill their own kind all the time. When Humans kill each other it's because they're listening to their animal instincts instead of their rational mind.
That's a major sweeping generalization. SOME animals kill their own kind all the time. There are plenty who do not.
Human Beings are the ONLY creatures who have committed genocide and wage war. Now, one could say that it has to do with sentience... that lower forms of life can't think of the organization required to do such acts. But gorillas have been known to team up to defend themselves against predators. If they wanted to wipe out an entire tribe of other gorillas, they would conduct their own kind of warfare. They do not.
Elephants will attack other creatures and even out of malice. There have been ones kept in captivity that aren't happy with their predicament and over time, building up enough anger about their situation until they eventually "pop" and go on a rampage, killing any human in sight (even those who had nothing to do with their captivity). But that's because the elephant was under duress. Elephants do have territories, but they don't seek to expand them with the waging of war against other elephants to achieve that kind of objective.
Humans conducting genocide aren't following primitive instincts. They are following a rationalization passed down to them from those in control of their group. And those who go on mass murder sprees are often mentally deranged. Nothing instinctual about it--they're simply deranged or have a warped sense of morality.