Google Custom Search ( CSE ) is free product. Its a great tool.
Its easy to setup, just like plugin and play, with in a two minutes you can finish the basic setup and can Add Google Custom Search Engine in your Website
You can Customize Google Custom Search Engine by using API, or from Google Custom Search Engine Admin panel.
but it have some limitations in customization. like Google brand appears on Search result. Display ads on Search Result
Alternate to Google Custom Search Engine
Algolia ( Free plan available ) , Algolia Community build Free Plugins for Several CMS, Which makes easy to Integrate Algolia in your Website . ( WordPress, Shopify, Magento, Jekyll, Zendesk )
Apache solr is open source enterprise search platform build on Apache Lucene
Amazon AWS CloudSearch Fully Managed, Cloud based Custom Search Engine ( for Mobile, Web App )
WordPress Plugins
Relevanssi replaces the standard WordPress Search Engine with extra features. 100,000 + Active Installation. one of the popular Search Engine plugin in WordPress. Relevanssi premium version
Search Everything Improves WordPress default Search functionality. 90,000 + Active Installations.
For most use cases Cloud based Search Engines are best instead on all the things loaded on your server.
Cloud Based Search Engines crawl, index your site on there server and manages. means less load on you server. And its easy to Setup, Maintain.
If you have large traffic website. Instead of using WordPress plugin which index you site on your Web Server use Cloud based Search Engine. ( for setup may need to use plugin, even if it is cloud based search engine )
But there are some Advantages of using WordPress based Search Engines
WordPress plugins Search Engines can know how your website content is, they know the popular products like Woo commerce, easydigitaldownloads. and know how they are structured. So more customize feature, better search result.
For Large Enterprises any thing is fine. Apache solr may be one of the best choose.