A&G have a note that carefully differentiates the double questions from a similar alternative—questions featuring the particles aut or vel / -ve. Let’s look at this distinction.
I ask whether he acted unjustly or dishonestly: quaerō num iniūstē aut improbē fēcerit.
I ask whether he acted unjustly or dishonestly: quaerō utrum iniūstē an improbē fēcerit.
In the first question, there are two options on the table, neither of which are necessarily true. It may be that he acted neither unjustly nor dishonestly. In the double question (the second example), it is clear that he either acted unjustly or dishonestly. We have to pick one.
I don’t really like A&G’s example, so here’s another:
I ask whether she likes cats or dogs: quaerō num felēs aut canēs amet.
I ask whether she like cats or dogs: quaerō utrum felēs an canēs amet.
In the first of these two questions, we know nothing about this girl. We’re merely curious about whether she like animals. We might expect our respondent to say something such as, ‘no, she likes birds.’ In the second example, we asking whether she’s a cat-person or a dog-person, assuming she’s either one of the other.
These two types of questions are identical in written English, and differentiable only in stress pattern. In the first question, we would stress ‘ask.’ In the second question, we would stress ‘cats’ and ‘dogs.’