Difference between last_name and user_lastname

Short answer to “Which property am I supposed to use“: Use first_name and last_name.

Longer answer:

The properties first_name, user_firstname, last_name and user_lastname are defined in the WP_User class, and despite the different names (i.e. one with user_ and the other without that prefix):

• Both first_name and user_firstname use the value of the first_name meta.
• Both last_name and user_lastname use the value of the last_name meta.

But user_firstname and user_lastname were the property names used in WordPress prior to version 2.0.0 (14 years ago..) — see get_the_author_firstname() and get_the_author_lastname() here and here, and that properties are still supported for backward compatibility, but we should just use the ones without the user_ prefix (e.g. first_name and not user_firstname).

// Both of these return the same value - the value of the meta named first_name.
var_dump(
    get_the_author_meta( 'user_firstname' ), // works
    get_the_author_meta( 'first_name' )      // but better
);

// Both of these return the same value - the value of the meta named first_name.
$current_user = wp_get_current_user();
var_dump(
    $current_user->user_firstname, // works
    $current_user->first_name      // but better/shorter...
);