Default Search of WordPress looks great and does a great job, but there are a few reasons that you should replace it with Google Custom Search Engine.
Here are the reasons:
Reduce Server Usage: When someone search using your default WordPress search function, your server receives the request and processes it every time to display search results. All this contributes a lot to the increased server usage. However, if you use Google Custom Search Engine, all the processing work is done by Google and you get the results.
Better Search Results: With Google Custom Search Engine your users get much better results because Google Custom Search Engine uses powerful algorithms to understand the query of the user. Let’s say your user search for a synonym of a keyword who related posts you have but the exact keyword is not present, then the default search engine will show that content is not available(404). However, when someone searches with the help of Google Custom Search Engine, he/she gets the result because Google also figures out that the synonym of this word is present. That’s why Google Custom Search Engine delivers better search results.
Monetization: I am not sure that you’re aware of this thing, but you can also use Google Custom Search Engine to make some extra bucks as revenue if you’re a Google Adsense user. Follow this guide to set up it for your blog,
Additional tip: If you’re using WP2Static or a static website, the only way to add a search feature is to use Google Custom Search Engine, as you see on GulshanKumar.net, if you want to know more about WP2Static read this guide.
If you have a decent knowledge of HTML and CSS, you can make your Google Custom Search Results look very similar to the default WordPress search result.