Elastic Search

It is required to make sure that all the necessary configurations were made.
First of all, please check what search engine is selected for your site in default Magento settings:

Stores > Configuration > Catalog > Catalog > Catalog Search > "Amasty Elasticsearch"

Secondly, check the general settings of the extension:

Stores > Configuration > Amasty Extensions > Elastic Search > Search Engine > "Amasty Elasticsearch"

Then, click "Test Connection" button and make sure that you see "Successful! Test again?" message.

If all of the above is set up correctly and search still gives no results, please create a support ticket with us as there might be a conflict with some third-party extension in your site.

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.