I agree that incorrect can be a "softer" word than wrong, particularly when pointing out someone's mistake. The two words indeed overlap in meaning, but the word wrong can be used to refer to moral lapses, while incorrect isn't really used that way. That's why wrong can seem like a "more negative," "more harsh," or "more personal" word than incorrect. Compare these definitions from Collins:
1 not correct or truthful ⇒ the wrong answer
2 acting or judging in error ⇒ you are wrong to think that
3 immoral; bad ⇒ it is wrong to cheat
1 false; wrong ⇒ an incorrect calculation
2 not fitting or proper ⇒ incorrect behaviour
As for any difference where you'd use one word for speech and the other for writing, I wouldn't agree with that difference. When talking about grammar problems, I think you could use either one, irrespective of whether speech or writing is being evaluated:
- That sentence uses the wrong verb tense.
- That sentence uses an incorrect verb tense.
In that case, wrong isn't all that more "harsh" than incorrect, because it's clear from the context that we are only talking about verb conjugation, and not some kind of moral belief or behavior. I think either sentence could be used to correct someone's speech or writing.