Edit: it occurs to me that DNA/RNA are made of natural symbols, which is a pretty good argument for symbols being important.
A good reason to use symbols in communication is for reliability (which is needed in the DNA). With a discrete set of symbols you have most likely no degradation (instead some recovery occurs -- like when you interpret a poorly written letter), and with a small chance you understand a completely different symbol. This stops error propagation.
Moreover, symbols allow for easy combination, which expands indefinitely the vocabulary with a fixed cost.
It doesn't seem like chimps are able to acquire that the same as humans are, given things like https://en.wikipedia.org/wiki/Nim_Chimpsky
The signs end up being things like "Grape eat Nim" or "Give orange me give eat orange me eat orange give me eat orange give me you."
You might think that things like this might reinforce the idea that language capacity is innate in humans, and that it differs in qualitative (and not merely quantitative) ways from communication in other species, but people are still wrapped up in some combination of strict behaviourism and the tabula rasa on the one hand and cultural constructionism on the other hand that they can't seem to see the data sitting in front of them. I don't think we'll get very far until we accept that the communicative aspect of language is merely a lossy format for serializing and transmitting internal data structures that may or may not have any correspondence with physical realities (apart from neuronal state). "A language" is just an agreed-upon format, like JSON, XML, or Gerry Sussman's "everything is a Lisp" notation(s), but without that config file, the general-purpose serializer/deserializer library is just unreachable code.
[1] https://en.wikipedia.org/wiki/Genie_(feral_child) [2] https://en.wikipedia.org/wiki/Nicaraguan_Sign_Language