That's not quite right. First of all, humans are primates, make no mistake about that. Secondly, other species of primates
do have the brain structures which allow for language. Many studies have suggested that some of the great apes (gorillas, chimpanzees, orangutans, etc.) naturally use language to communicate with one another. What they do lack are the necessary modifications to the throat and mouth which are required for speech. But, they can use grunts and gestures to effectively communicate with one another.
Other studies have shown that gorillas and chimpanzees can learn language, and apply it to complex concepts, much the same way human children do. They've been shown to have a sense of "self", just like we do, and we're more similar to them than we are different. For some good information on primate language studies, read the articles
here and
here.