In order to discuss the topic of Primates and language, the best place to start is probably with a basic definition of language and communication so we understand the difference.   To poorly quote a saying I once heard, "history is written by the victors" and with that in mind, so are the rules.  So bearing with that thought, if this were the "Planet of the Apes" these definitions might be different.  In the Hominidae family however, it is us, the Homo sapiens, that write these rules.

*language \Lan"guage\, n. [OE. langage, F. langage, fr. L. lingua the tongue, hence speech, language; akin to E. tongue. See Tongue, cf. Lingual.]

1. Any means of conveying or communicating ideas; specifically, human speech; the expression of ideas by the voice; sounds, expressive of thought, articulated by the organs of the throat and mouth.

*communication \Com*mu`ni*ca"tion\, n. [L. communicatio.]

1. The act or fact of communicating; as, communication of smallpox; communication of a secret.

2. Intercourse by words, letters, or messages; interchange of thoughts or opinions, by conference or other means; conference; correspondence.

     Now, looking over these definitions on a basic level, I am already presented with questions.  First, "Any means of conveying or communicating ideas" strikes me as an open ended statement.  Even my cat can tell me when she's hungry or needs to go outside.  To read further into that definition only strengthens this thought.

     Communication seems to be an easier issue.   The act of conveying thoughts or opinions by some means.  In a sense, if you communicate, you use language of some format.  But the issue isn't if Primates use language or communicate.  No, if that were the question, on that facet we would fail.  So, to make the issue complicated, we first argue that only humans use language, and then qualify it by stating that Primates don't use 'human' language.  Hum, wonder why they call it human language? 

     But let's step back, and revel in the human tongue for a moment.  I am aware scholars spend their entire lives defining what language is.  In fact, one of the advantages of being human, sarcasm aside(?), is our unique ability to spend a millennia in discussion only to find no resolve.  However, when it comes down to, dare I say, trying to show that we, humans, are not the only ones that can use language, boy do the solid definitions seem to come pouring in.

* Definitions taken from www.dictionary.com, but any standard dictionary should contain similar references. 
Similar definitions were found at  www.m-w.com.