I came across this sentence,
I don't know, nor do I care.
I didn't know you can use nor in this way.
What is the difference between that sentence and this one:
He can't see nor hear.
I came across this sentence,
I don't know, nor do I care.
I didn't know you can use nor in this way.
What is the difference between that sentence and this one:
He can't see nor hear.
The "nor" here means "and ... not". To paraphrase the first sentence you quote:
I don't know, and I don't care.
Note the word order after "nor": the verb comes next, followed by the subject.
Your second sentence has only one negative (in "can't"). "Can't" governs the coordination of two infinitives, "see" and "hear" which are both positive. "Nor" isn't necessary here; "or" would do. As for why the writer used "nor", they might have just wanted to make the negation more emphatic, or they might have thought of a sentence such as
He can't see, nor can he hear.
and then shortened it by removing the second "can" and "he" as they just repeat the first "He can".
There is no difference in meaning; the only difference is grammatical and in the frequency of usage. In the second sentence the subject and the verb are ellipted. Using them yields a sentence that is correct; however, in order to do so you have to use a "do construction" for plain verbal forms and invert the subject and the auxiliary for modal forms.
The ellipted form is not found much (rare) in the case of plain verbal forms; this ngram tends to shows that because in the graph for "I do not know nor" are included second conjuncts with ellipsis of the verb only and also those cases when the second verb is followed by a complement, which makes the structure usual; as well it covers the forms "…nor can…", "…nor did…", etc.
It does not snow nor rain all day as in certain parts of northern France.
(1857) I do not know nor believe that anything was offered or paid by anybody to us , or to any other person […] (ref.)
(2007) I do not know , nor want to know , the whys and the wherefores. (ref.)
(2010) I do not know nor wish to know what destiny […] (ref.)
Rare cases of the use of the verb without complement and without do support
(2011) In this I may not be unique, it may be thus for everyone, more or less, I do not know, nor care. (ref.)
(2012) And what was that first Golden Fleece? I do not know, nor care.(ref.)
(2009) I do not know nor care. This hideous giant travels deep underground, wearing mud and stones as clothes, and calls itself a graveyard, a place for all to be eaten by worms...(ref.)
(1987) Now about Duke and Larry: I don't know nor care. Whenever anyone comes here to live, I make it plain that this is ... (ref.)
(1872) Of course it will be a boy , " on it , a six - weeks baby , so she said ; perhaps said I. she did; I don't know, nor care. (ref.)
(2001) and they prayed to things which can not see, nor hear, nor walk; (ref.)
Often enough, a comma is used before "nor" and sometimes even a full stop.