As I said in another comment, one of the defining characteristics of the autism spectrum is a blunted sense of empathy. As you say, that blunted empathy can mean that the autistic person doesn't hear the emotional content, reacts to it inappropriately, or is not able to effectively communicate emotional content themselves.
Come to think of it, if people on the spectrum aren't communicating emotional content, or are doing it very poorly, that might explain part of why some autistic people think they're communicating precisely with carefully chosen words, but their intent and meaning is still being misunderstood.