I'd like to know whether I should ask like "Why don't you do sth?" or "Why do you not do sth?"
Are both correct? Does each differ from each other?
Here we still hear both, though the first is more common.
(1) You have asked a very interesting question.
(2) In a book entitled The Grammar Book, the authors say that
sometimes there is a little difference:
Why didn't you do the work? The authors say that sometimes it
could express a negative sense. Something like: You should have
done the work. That's what you get paid for!!! So do it now!!!
Why did you not do the work? The authors say that this way
is not only more formal but also "more neutral." For example
(only my example -- not the authors'): Excuse me. But why did you
not do the work? Were you sick? Worried by family problems? Maybe
there is a reason for your not doing the work. Could you please tell me?