What is the difference between Elastic Search and Advanced Search extensions for Magento 2?

The principal difference is that the two extensions use different search engines. Advanced Search relies on the default Magento MySQL search engine, when Magento 2 Elastic Search uses the up-to-date elastic engine which actually influences the search process in general. Using elastic engine in your Magento will allow you to get better speed, get good performance even with large product quantities and the ability to use wildcards, spell correction and different search logic, easily configuring it all for each product attribute. Having Elastic Search in your site would be more advantageous as all of the useful features of Advanced Search are there with better performance.