They are actually quite similar. AvPD is often seen as a more severe version of SAD, and I think there are lots of people who have both.

I read somewhere, though of course that was just one website's opinion, that the difference is that people with SAD see it as a problem outside of themselves that they can overcome, whereas people with AvPD feel it is a part of who they are.

It's not true that avoidants don't want to be around peope at all. They do long for connection. There is just that fear of rejection and feeling less than others that makes it so hard.