I discovered the problem. The discounted prices display Ok in the store frontend. The problem was that I was developing a script "external" to Magento (thus not a Magento module), something like:
<?php
set_time_limit(0);
ignore_user_abort();
error_reporting(E_ALL^E_NOTICE);
header("Content-Type: text/plain; charset=utf-8");
require_once "app/Mage.php";
// Get default store code
$default_store = Mage::app()->getStore();
...
For everything to work properly it seems that one must follow the proper Magento bootstrap, and develop everything as a module. My script was so simple that I thought it wouldn't be necessary to code a complete module. In other words, everything in Magento should really be a module.
Concluding, using the module approach, all the methods work as expected:
$_product->getPrice()
$_product->getFinalPrice()
$_product->getSpecialPrice()
Thank you all for your input.