In what situation would you use 'so' instead of very? For example, I'm very sorry. I'm so sorry. (When you apologize to someone)
Does 'so' sound natural in the context? Is he an old man? Yes, he is so old. How old is he? He is seventy years old.
There is a formality distinction between so and very. Using so connotes a degree of familiarity, warmth, surprise, exclaim, and/or incredibility that isn't necessarily present with very.
For example, to say "He is very old." is to note that the person is older than the age that comes to mind then one thinks of old. The word very is generally a neutral intensifier.
However, one would exclaim "She's so old!" if they had just learned the subject's age and was surprised by the answer, or to underscore the subject's advanced age. This would be said in an informal setting, e.g. with friends, and not normally in formal settings, e.g. in writing, to people of authority, or at a black-tie event.
In your other example: I'm very sorry. and I'm so sorry! have similar meanings but differ in formality. One would use the former to express condolences at the funeral of an acquaintance; the latter I use when telling a friend I am running late.
I explain the difference between so and very like this:
In formal English, I use so to describe exactly how happy / sad / dirty / etc something is:
When I'm being more casual, I don't need to describe exactly how happy / sad / dirty / etc it is. I will ask you to imagine what I might say:
In one sentence, so means: I could say a lot more about how happy / sad / dirty / etc it was, but I don't have time now.
So is stronger than very. Of this use of so, the Oxford English Dictionary comments
A typical citation is this from ‘Pickwick Papers’ by Charles Dickens: ‘My dear brother is so good.’
So is used instead of very when we want to emphasise the adjective or the adverb following it. In written form, I'd expect to see an exclamation mark at the end of "I'm so sorry!" or "I'm so tired!"
In the second example you write, "He is so old" doesn't sound natural unless the sentence is continued with something like: "He is so old that he can barely walk." If the sentence doesn't continue, I'd use very.
I would submit that very has become the mildest of intensifiers. To use it is tantamount to "damning with faint praise." I think this is precisely because very has no real quantitative value, yet it purports to suggest more than the bare adjective it modifies.
In fact, it brings the assertion of the adjective to earth, curtailing its force.
Now, so, on the other hand, has moved into the forefront of the intensifiers in spite of the fact that it too has no real quantitative value. The reason for its power is that it invites the listener to complete the comparative.
In fact, this is such a powerful adverb that the current vernacular has begun to use it in the service of modifying entire phrases.
In each of these cases, the speaker is making a very emphatic statement — er, I mean, the speaker is so making an emphatic statement.
Now, to the point about so very — this much stronger and more heartfelt than saying very sorry, and is used on occasions when whatever offense or tragedy has occurred is so grievous that the amount of sorry cannot be measured, and in fact any attempt at quantification would be to trivialize the situation.
So and very differ in the degree of expression.
"I was very tired that afternoon, that I had to avoid the meeting"
Means you were tired to a greater extent.
"I was so tired that afternoon, that I had to avoid the meeting"
Means you are tired beyond the limit. "So" can be used to press "very". I mean the next stage of "Very".
I am so happy to win the match.
I am very happy that I got a rest for a day.