Learn English – I’m sorry vs i am sorry

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.

"Sorry" was originally a more general word for sadness or misery. You don't hear it used this way very much anymore, but its use in apology and sympathy both stem from that original sense. As an apology, it's essentially short for "I'm sad that I did this thing." As an expression of sympathy, it's short for "I'm sad for you, as you go through this situation."