You'd think that the act of "smiling" (twisting the corners of your mouths while baring your teeth) conveys a state of happiness, like it works in humans. You'd see a monkey "smile" and think: That monkey is happy.
But in most other primates, the act of "smiling" is an act that's likely to be followed by violence, so it's actually seen as a threat. So if you're in the wild, see a monkey smile and think "oh he's happy, I'd better smile back" he might assume you're "threatening back" and attack you.
And why do you think that monkeys cry like humans? We are the only primates that shed tears as a sign of emotional distress (source: http://www.scientificamerican.com/article.cfm?id=are-humans-...).
Also, tears are not necessary to cry. In your source, scientists prefer to conservatively talk about "distress vocalizations" instead of crying. But that's necessary because they are so similar in the first place! That's my point: they express their distress! In everyday life (not writing a paper), I'd just say that "this ape is crying because her baby died". Heck, even my dog, I'd say that he's crying. I don't care that there are no tears. They have a mammalian limbic system, and the rewards it contextualizes have to be evolutionarily sensible!
Btw, and I know that's not a strong argument, but my eyes don't produce tears (following some medication). Does it downgrade my crying into a distress vocalization? If not, is it because you think that you know how I am feeling? How I am thinking? really? How do you know that?
PS: as a_bonobo yourself, you know that what I say makes sense ;).
I'd also like to point that in the case of great apes and some other mammalians, the brain structure is not very different than a human brain. We can reasonably expect that a similar emotional state is acheived when a similar external display is observed. I mean that for instance, when they cry, we can assume that they are sad, because their brain is quite similar to ours.