Normally, when the and is used in this kind of construction, it implies a combined activity. Two things happen either simultaneously or very close in time to each other.
Don't drink and drive [at the same time].
I added the text in brackets to make the understanding of the conjunction clearer. This means don't drive immediately after you've been drinking. (Or, in theory, it could also mean don't drink while you're driving.)
If somebody wanted to express the idea that somebody should not drink and also that they should not drive, and in a way where those two things don't have some kind of intrinsic connection to each other, then they would say:
Don't drink or drive.
If and were to be used, the commonly assumed meaning of one thing in relation to the other could be avoided if the verb were not omitted in the second part of the parallel construction:
Don't drink and don't drive.
This explicitly avoids the implication that drinking and driving is a single action (or concept), making it clear that it's two separate things.
So, in your example:
He can't read or write.
This sounds normal. He can neither read nor write.
If the or were to be replaced by and, while still preserving the sense of two completely separate activities, the following would be used:
He can't read and he can't write.
But failing to repeat the verb results in a different interpretation:
He can't read and write [at the same time].
If the final version really were what was intended, it would likely be phrased a bit differently:
He can't read at the same time that he's writing.