There is a mountain of evidence to suggest that human speech is evolutionarily rooted in the gestures of previous primate iterations.
Gesture/language researcher Luciano Fadiga called spoken language "“...the evolutionary refinement of an implicit communication system already present in lower primates, based on a set of hand/mouth goal-directed action representations.”
For instance, it is believed that human language centers Broca's area (involved in speech production) and Wernicke's area (involved in speech understanding) evolved out of the part(s) of earlier primate brains that handled gesture production/understanding. Here is one picture of two of the proposed homolog sites in several primates, including the human (homo):
...and yet other primates cannot talk. Aside from lacking the more advanced language brain structures of humans, at least some part of the problem may lie in physiological differences, specifically in areas like the voice box, mouth, and tongue.
Here is a neat video of the famous gorilla Koko, who knows upwards 1,000 sign language signs, and can understand about 2,000 English words:
Communicative Signaling Activates ‘Broca's’ Homolog in Chimpanzees
This 2008 study by Tagliatela et al. showed activation of numerous language homologs in the brain (including Broca's) in gesturing chimpanzees. In the study, which included 3 adult chimpanzees, a researcher placed a small piece of food in front of an isolated chimp's cage, so as to be in view of the chimp, but well out of its reach. After two minutes, the researcher would verbally acknowledge the chimp, and present it with the morsel of food. The researcher then departed for two minutes. At the end of this time period, the researcher returned and started a process over. This portion of the experiment lasted 40 minutes, at which time the chimp's brain was evaluated using a PET scanner.
The results showed portions of the inferior frontal gyrus to be very active when the chimps produced their communicative gestures. This significant IFG activity suggests that there are precursors to the language production of humans. In this scan of one chimp's brain, note the various areas of activity, including a homolog for Broca's area (bottom right image, temporal lobe area):