Depending on how you convert synapse count to parameters, the brain also has something like a thousand trillion parameters. In that light it's pretty darn surprising that an artificial neural network can produce anything like coherent text.
I believe the answer lies in how "quickly" (and how?) we are able to learn, and then generalize those learnings as well. As of now, these models need millions (at least) examples to learn, and are still not capable of generalizing the learnings to other domains. Human brains hardly need a few, and then, they generalize those pretty well.