"I have" and "I've got" both mean possession. Also, "I have to" and "I've got to" both mean obligation. When we use "got", it's always with a contraction: "I've got". "I - have - got..." is not natural. In questions and negatives, we use "Have you got...?" and "I haven't got..." "Have" can be a main verb or a helping verb. The helping verb functions differently, compared to a main verb.

The word "got" shows ownership of the books or possession of the books. Or in this case a lack of ownership as the statement says: I own zero books. You might hear someone say "I haven't got any books" when they are annoyed or angry but it could be used when someone is not angry also. It would be best to say. "I don't have any books" which means exactly the same thing.

Muchas Gracias Scott,

"Got" es el pasado participio de get como "obtenido" en español. Así que tenemos "No he obtenido libros" o en inglés "I haven't got any books." En EEUU se dice "I haven't gotten any books."

Muchas gracias Izaya :)