Kim A. Bard in Scientific American:
This is an interesting question and one that is very well phrased. Humans are primates and it is important to consider our behaviour within the evolutionary context of other primates. But it is also necessary to define what we mean by crying. If crying is defined as the act of tears coming from the eyes, then simply, the answer is yes: tears appear to be unique to humans among the primates. The more interesting and complex answer, however, concerns what crying can mean in terms of emotion, emotional expression, and/or feelings.
We can think of crying in two ways: as an emotional expression, with or without feelings (for example, of sadness, distress, or pain), or as a communicative signal (for example, of vigorous health of babies, or with communicative intent to invite caregiving and/or solace). In terms of emotional expression, crying might include any or all behavioral indices of distress (such as vocalizations, body movements, and facial expressions), or indices of sadness (for example, depressed body postures including slumping shoulders), or of pain. Crying as an emotional expression has been used to describe the vocalizations of many primates, including the coo vocalizations of squirrel monkeys and the whimpers and screams of chimpanzees. Crying has also been used to describe the vocalizations of monkey and ape infants when they are being weaned, and when they are separated from their mothers (either temporarily due to losing sight of the mother or permanently due to maternal death).