In my accent all of these words have an identical vowel sound:
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Sorry:
Caught
Cot
Hot
Don
Dawn
On
Lawn
Mall
Call
Ball
Tall
Ad infinitum.
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Interesting! They do not, to me. I’ll group the ones whose vowels are the same:
Cot
Hot
Don*
On*
- There’s a slight difference between how I say Don and on vs. cot and hot, but it’s there.
–
Caught
Dawn
Lawn
–
Mall
Call
Ball
Tall
In New York, Don and Dawn are two completely different people.
Baffling, isn’t it? To think they don’t rhyme makes me shake my head.
That’s what we call “accents.”
That’s what it’s all aboot, eh?
(And no, we don’t really say “aboot.”) 
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Okay now what about Erin and Aaron?
I have a slight difference in my pronunciation of these two names, but I know most people in this neck of the woods don’t.
I golfed with a guy a couple of years ago and was introduced to his son “Erin.”
Well, that’s odd I thought, untill I saw the score card with “Aaron” printed on it!
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These are indistinguishable in much of North America because of Mary-merry-marry merger.
To distinguish them, “Aaron” has the “cat” vowel [æɹən] and “Erin” has the “egg” vowel [ɛɹɪn].
Back to Barb and Bob.
I live in NY state. Some years ago I sang in a choral group. One of my fellow tenors for a spell was a person with white hair and glasses who was either
–a very slightly built man with a relatively high voice and no facial hair whatsoever, named Bob
or
–a woman with a low voice and an extremely boyish figure, named Barb.
I heard at least three people, including Bob/Barb him/herself, say the name, and I could not identify which was being said. You’ll understand when I say I didn’t quite feel comfortable asking…
I say the names very differently, but these folks did not. (I grew up in the Midwest, not here.) the person in question didn’t sing with us for long…about three years later I did see a picture in the local paper with a caption beginning “Barb…” so I guess that finally settled the question.
In the 80s, when DOS was a new word, an Apple manual helpfully said it was pronounced like “toss.” I called Cupertino to ask if it was Dahss, as in Californian, or Dawss, in Midwestern. She prettily confirmed the former. I wish they all could be California girls.
I can never tell if I just have a lazy ear or if I actually grew up with an accent that totally coincidentally (I was raised in west-central PA) merges every one of the above vowels into a single sound. The merry/mary/marry thing, the cot/caught thing, etc…every one of these things sound exactly the same within the comparative groups to me.
ETA: And I just realized I’ve also had hearing issues throughout my life…many ear infections as a child, a tendency to retain fluid in the middle ear throughout my life, etc, may very well have hindered my ability to distinguish vowel differences. It’s never been a profound impairment…I’ve never been literally deaf. But I spent the majority of 46 years with eardrums that had very little flexibility and movement because of retained fluid in my middle ear. It’s just been something that I’ve been dealing with so long that it doesn’t spring to mind in discussions like this because to me it’s NORMAL.
When I moved from Ohio to NYC, I had to learn to distinguish among Mary, merry and marry. Then, 25 years later, when I relocated back, I had to re-learn to pronounce them identically. And the soda/pop thing. But I never pronounced cot and caught (or Don and Dawn) the same, regardless of where I lived.
A kid I went to high school with also moved to NYC, where he became a successful lawyer. I saw him on tv once, and lo and behold, he spoke like a New Yawker. Some people are more flexible than others.
In Boston, at least, Barb = Bob and Bob = Bawb. Easy-peasy! 