Snippet Plugin Not Working

Quite possibly because snippets run at an earlier point in the WordPress load order then functions.php code. Try wrapping your remove_filter call in a function:

add_action( 'init', 'wpbe_wpmandrill_compatibility_init' );

function wpbe_wpmandrill_compatibility_init() {
	remove_filter( 'mandrill_payload', array($wp_better_emails, 'wpmandrill_compatibility') );

	// do your own email processing
	add_filter( 'mandrill_payload', 'wpbe_wpmandrill_compatibility' );
}

function wpbe_wpmandrill_compatibility( $message ) {

    global $wp_better_emails;

    // make sure you have the %content% tag in the template
    if ( $wp_better_emails->check_template() ) {
        // clean < and > around text links in WP 3.1
        $message['html'] = $wp_better_emails->esc_textlinks( $message['html'] );
        // make links out of URLs
        $message['html'] = make_clickable( $message['html'] );
        // set template
        $message['html'] = $wp_better_emails->set_email_template( $message['html'] );
        // replace variables
        $message['html'] = apply_filters( 'wpbe_html_body', $wp_better_emails->template_vars_replacement( $message['html'] ) );
    }

    return $message;
}

I forgot to add global $wp_better_emails; to the first function. Try this code instead:

add_action( 'init', 'wpbe_wpmandrill_compatibility_init' );

function wpbe_wpmandrill_compatibility_init() {
	global $wp_better_emails;

	remove_filter( 'mandrill_payload', array($wp_better_emails, 'wpmandrill_compatibility') );

	// do your own email processing
	add_filter( 'mandrill_payload', 'wpbe_wpmandrill_compatibility' );
}

function wpbe_wpmandrill_compatibility( $message ) {

    global $wp_better_emails;

    // make sure you have the %content% tag in the template
    if ( $wp_better_emails->check_template() ) {
        // clean < and > around text links in WP 3.1
        $message['html'] = $wp_better_emails->esc_textlinks( $message['html'] );
        // make links out of URLs
        $message['html'] = make_clickable( $message['html'] );
        // set template
        $message['html'] = $wp_better_emails->set_email_template( $message['html'] );
        // replace variables
        $message['html'] = apply_filters( 'wpbe_html_body', $wp_better_emails->template_vars_replacement( $message['html'] ) );
    }

    return $message;
}

Also, make sure that you don’t have this snippet in your functions file as well as in the Code Snippets area, just in case.