These two words appear to mean exactly the same thing, come from the same root words, and can be used interchangeably in every situation I can think of. How do two such nigh-identical words happen to make it into the dictionaries? Is there a nuance here that I’m missing?
and don’t forget bewilder… although that must come from a different root.
In my mind “astound” has nuances of stun and dumbfound, sort of astonished in a hit-in-the-head-by-a-2x4 sort of way. “Astonish” is more “heavily surprised.”
This is meant as a native-speaker datapoint, not a definitive answer, and I know that there’s nothing in the etymologies to support me.
FWIW, I (also without cite) concur with this usage; when astounded, I am deeply but quietly perplexed; when astonished, I gasp (or vocalise) in amazement.
See? The plural of anecdote sometimes is data.
I also concur.
I think the difference can be seen as well. When “astounded” I narrow my eyes and bring the eyebrows down; when “astonished” my eyes go wide and my eyebrows go up.
Subtle? Sure. Unscientific? Absolutely. But fun!
Similar input here – for me, “astounded” involves my head moving backwards a bit, while “astonished” is my eyes opening and eyebrows moving up.
As a writer and editor, I’d use “astounded” in circumstances where “stunned” might work, but “astonished” where “surprised” is more appropriate. YMMV.
English is an extremely rich and expressive language, and is loaded with examples of words that mean the same thing, or almost the same thing, or maybe the same thing to some people but different things to other people, or different things at different times. This sort of thing just happens with natural language. Why say “amazed” when you’ve already got “astonished” and “astounded”? It’s just how English works.
I agree.
I can be **perplexed, dumbfounded, flabbergasted, incredulous, overwhelmed, speechless, bewildered, gobsmacked ** and totally surprised.
But not if I have my trusty thesauras!
Spelling?
How about using surprise instead as it means to cause to feel wonder, astonishment, or amazement, as at something unanticipated.
And bewildered for me is more like a crumpling in of the face, with perhaps one eyebrow lifted.
Likewise here, with the addition of a head tilt to the right.
Astonishment is a far more immediate reaction than being astounded. You pass through astonishment to astounded. Astoundedness lingers whereas astonishment may pass. Bewiliderment is a near-permanent state of astoundedness.
Perplexedness is the process of rationalising astoundedness. If it fails, you become bewildered.
mm
Amusing anecdote related to the OP: John W. Campbell, Jr., the quite famous editor of what was then Astounding Science Fiction (now Analog), was noted for coming up with plot ideas, which he would then have his circle of authors write up as stories. It should be mentioned in passing that one of Astounding’s competitors was Amazing Stories. There’s a famous account of him spinning off the basis for a story to a visiting writer, who commented, “That’s an amazing idea!” Campbell smiled, and shook his head. “No,” he responded, “it’s an Astounding idea.” 
I would argue the reverse for the two a-words: One can continue to be astonished at something, but one can’t continue to be astounded. I do, however, agree on bewilderment: I (and my advisor) am bewildered by a problem I’ve been working on for some months, and I remain bewildered by it as time passes.