Can I run a second wordpress site as a subdomain without using multisite?

Point the subdomain to the same directory as the main site, and define different settings in your wp-config.php per $_SERVER['HTTP_HOST']:

Example from my local setup:

switch ( $_SERVER['HTTP_HOST'] )
{
    case 'zzl.dev':
        $table_prefix = 'zzl_';
        $table_name="zzl";
        break;

    case 'wpbuch.dev':
        $table_prefix = 'wpbuch_';
        $table_name="wpbuch";
        break;

    default:
        $table_prefix  = 'wp_';
        $table_name="wpdev";
        break;
}

$sub                 = '/wp-content/' . $_SERVER['HTTP_HOST'];
const WP_CONTENT_DIR = __DIR__ . $sub;
const WP_CONTENT_URL = 'http://' . $_SERVER['HTTP_HOST'] . $sub;
const DB_NAME        = $table_name;

You can change much more variables in the switch: all DB_* definitions, WP_PLUGIN_DIR and WP_PLUGIN_URL (to share the plugin list between different sites), WPLANG and so on.