JP, there IS a connection between several different kinds of mutisms and autism. This is one of my "special interest areas", since I had problems and everybody had questions and nobody had answers...I had the time and a little research background to go find some answers. So I could probably overwhelm you with information on the subject.
I will try to answer your original question. The connection between selective mutism and autism is potentially in a couple of different areas. One area is in the cerebellum part of the brain, which is sometimes malformed in people with ASDs and that makes for odd wiring for coordination of all sorts of things, and communication is one of them. I have a very mild form of that. Another issue is that the left temporal lobe doesn't exactly come pre-programmed WITH speech, but it does normally come pre-programmed to LEARN speech a certain way in NT's as toddlers. They still do not understand if it's a lack of the pre-programming or what, but when speech is not learned in the normal way some form of mutisms often result, and the milder ones may not even affect you very much until later in life, or when you are under a lot of stress. Or, as a professional once put it to me, maybe I was supposed to be mute, or partially mute, and I was smart enough to "hotwire" my brain for speech, with some help from my determined mom. In that case speech would be a doubled edged sword, because I'm constantly pushing myself too hard to use it and causing myself problems and frustrated with it...but oh how good it is to have the world available to me that speech has made possible.
When we have a high enough IQ to "hotwire" our brains for speech, and learn something we lacked the pre-programming for just because we were smart enough to figure it out...well, that's potentially an HFA/Asperger's thing. But that doesn't mean that speech is going to be as easy as it is for other people. The more stress you are under the more difficult it will become to juggle all the bits that we have to learn that don't come naturally...tone, volume, facial language, body language, etc. When you get tired it'll be more difficult. Anxiety would do it too. Some meds can also affect it for some reason.
EEG's of mutes show low/slow left temporal lobe activity. When I have had 2 episodes show up on EEG's they also showed up as slow left temporal waves, but I gather it's not a constant thing with me, and that's what makes the mutism selective. Elective mutism is when you choose not to speak.