"number of books" or "book count"?

Solution 1:

Your second sentence is much more formal and stilted; you wouldn't use it in conversation unless you were doing a count of books. Count here means:

The result of a tally that reveals the number of items in a set; a quantity counted.

Your first example is more "natural" but not as "natural" as "There are nine books." (In conversation, I would probably even say the ungrammatical "There's nine books.")

Solution 2:

This is strictly a style question. There's no context, so there's no way to tell which is more "natural". Both are fine in appropriate contexts, both are idiomatic, and both are typical native speaker English.

There's no subtle difference between them. One has more words and characters than the other. That's an obvious difference. If you're answering a question about how many books are on the table (How many books are on the table?), neither is a good choice. You'd have to say:

Nine

or

There are nine

or

There are nine books

or

There are nine books on the table.

The differences between these four answers are both obvious and subtle: the fourth answer means you're in an EFL or other type of English class and the teacher's required you to answer in a full sentence that repeats most of what's in the question; the first answer means that you're a normal native speaker of English who understands that the only response necessary is this one word: "Nine"; the second answer means that you're just a tad verbose; and the third answer means that you're pedantic or patronizing because you think that the questioner has forgotten what he or she's asked about.