In what situation would you use 'so' instead of very?

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:

• I was so happy I could cry!
• He was so sad he felt like killing himself.
• It was so dirty I couldn't see what its real color was.
• etc.

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:

• I was so happy!
• He was soooo sad!
• It was so dirty!
• etc.

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 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.

He was rich.
He was very rich.
(Richer than rich? By how much? We don't know.)

The cat was black.
The cat was very black.
(Black is black. Very black is still black.)

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.

"It was so cold that day."
(How cold was it? Use your imagination.)

In fact, this is such a powerful adverb that the current vernacular has begun to use it in the service of modifying entire phrases.

I am so not going to do that.
We are so not going to buy anything from that store ever again.
I am so going to kick his little ass for him when he gets home.

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.