The Others are the rival tribe; the man-apes who scream at Moon-Watchers man-apes down by the river every day. The Others and the man-apes don't fight most of the time, they just shout at each other:
Though the man-apes often fought and wrestled one another, their disputes very seldom resulted in serious injuries. (1.15)
The conflict with the Others is, then, a kind of cold war. There's a wary animosity, but not open, effective violence.
Arthur C. Clarke was himself writing during the Cold War, the long period after World War II when the United States and Russia were engaged in a global conflict short of actual fighting. So the screaming between the Others and the man-apes is in part a symbol of, or an echo of, the Cold War. Humans, the novel seems to suggest, are always fighting and grappling with each other—and technology (whether a club or nuclear weapons) has the potential to make that conflict a lot bloodier and more dangerous.
The novel seems excited about how Moon-Watcher's use of tools has allowed him to win the cold war, bash in the skulls of the others, and become "master of the world." (5.11) At the same time, though, the book recognizes that technologies of death can lead bad places—nuclear weapons appear for a brief cameo at the very end of the book.
So the Others both show how you have to break a few heads if you're going to evolve, and suggest that maybe breaking a few heads is a sign that things aren't evolving so much after all, if you're still fighting one Other or another several millennia in the future.