University of Iowa paleoanthropologist Robert Franciscus answers:
The first question is whether Neanderthals were physically capable of speaking. That ability is tied to a few key anatomical structures, including densely packed nerves in the lips and the tongue and a voice box positioned low in the throat. Based on sketchy fossil evidence, some anthropologists argue that Neanderthals could make limited vocalizations but that they lacked the full range of modern humans; in particular, they were probably limited in the vowel sounds they could produce.
Perhaps a more meaningful question is whether Neanderthals were mentally capable of abstract expression. Symbolic behavior, such as art and body ornamentation, is thought to be closely linked to the sharing of complex ideas and spoken language. Neanderthal remains are occasionally associated with such symbolic artifacts, but those pale in comparison with the artifacts produced by early modern humans, suggesting a significant gap in linguistic abilities. Still, there is some evidence that Neanderthals and modern humans coexisted, shared cultural information, and even interbred—whether they spoke to each other or not.