Learn English – Any books/book you want

Both can be correct, depending on the words. You may expect more than one difference. For example:

What is the difference between a "smuggler" and a "pirate"?

(I assume that these words are very similar, and there is only a small difference.)

What are the differences between a "smuggler" and a "pirate"?

(I assume that these are different enough that there are multiple differences.)

It doesn't necessarily matter how many things are actually different, only how many you expect.

There is no general rule, if singular or plural should be used after no/any. It depends on the situation and context.

For your first example the correct sentence would be:

I couldn't find any "apples" in the basket.

Edit: Because any is used in front of nouns to refer to indefinite or unknown quantities or an unlimited entity, all countable nouns (e.g. apples, chairs, etc.) are in plural and all uncountable nouns are in the "singular" (e.g. money, bread, etc.).

For your second example, either sentence can be correct, depending on the context.