The WooCommerce search only looks in the product title and description. The theme Ajax search is the same but it shows the results immediately.
To search by other attributes like SKU, categories, etc you have to use a plugin like https://wordpress.org/plugins/advanced-woo-search/
To use any plugin that replaces the default search box disable the Ajax search feature in Appearance > Customize > Header > Header Icons
Then, in the Advanced Woo Search settings enable Seamless Integration
That’s the basic configuration.
Then, you can check the search sources