Categories and Tags Conflict after Woocommerce Installation

I was able to resolve this issue by changing the rewrite slug from product-tag to product_tag when registering the taxonomy. It looks like WP does not like the hyphen and prefers the underscore.

$args = array(
        'hierarchical'          => false,
        'labels'                => $labels,
        'show_ui'               => true,
        'show_admin_column'     => true,
        'update_count_callback' => '_update_post_term_count',
        'query_var'             => true,
        'rewrite'               => array( 'slug' => 'product_tag' ),
    );

register_taxonomy( 'product-tags', 'product', $args );