What is the difference between "Does she have the book?" and "Has she the book?"
In U.S. English, the difference is that "Does she have the book?" is normal, whereas *"Has she the book?" is not.
I can't speak with confidence about U.K. English, but I believe that in U.K. English, both forms are correct, and with no difference in meaning, but with "Has she the book?" being more formal and "Does she have the book?" being more informal.
Questions with to have are a special thing, because you can find three possibilities:
a) Do you have the book? The standard variant in AmE.
b) Have you got the book? - Mostly colloquial BrE. See remark below.
And sometimes you find
c) Have you the book? - This is older English or colloquial English in some regions. I would not use this variant as it really is limited to certain regions.
In my view the most logic variant is a). "to have" has the meaning "to possess" so it is a full/normal verb and the use of "to do" in a question or negation is the logic thing.
Remark In BrE the question and negation with to do is accepted. Influenced by AmE this is used more and more, especially among the younger generation. Examples: - Do they have a nice little house? - No, they don't have a house. - Did they have a nice little house? - No, they didn't have a house.
The question with simple inversion is common in special uses of to have such as - to have a sister/brother: Have you brothers and sisters?
and also in uses where to have has the meaning of there is/there are: - How many days has September? - Has the room (got) two or three windows?
First of all, I want to credit @F.E. for misleading me into to finding the answer to this question in the "Comprehensive Grammar of the English Language" by Quirk et al..
Quirk classifies verbs into:
Full / lexical verbs such as
Primary verbs: be, have, do
Modal auxiliary verbs such as
The property that classifies
have as a primary verb is its ability to act in a sentence as either a main or auxiliary verb.
Quirk discusses the uses of
have in section 3.33, 3.34 and 3.35. From his discussion, I extract here the following uses of
have acting as a main verb and combining with
do in the usual manner:
We don't have any money
Do you have a lighter?
and the following uses of
have acting as an auxiliary verb (acting as an operator in Quirk's words):
We haven't any money
Have you a lighter?
Quirk affirms that although this construction is traditional in British English, now is somewhat uncommon, particularly in the past tense:
Had she any news?
To further support that
Has she the book? is acting as an auxiliary verb, @F.E. gives in the comments an excerpt from the "The Cambridge Grammar of the English Language":
While dynamic have is invariably a lexical verb, stative have can behave as either a lexical verb or, in some varieties, an auxiliary. This means that for the negative we have either don't have or haven't (or the analytic forms with not), and analogously with inversion.