The advantage of custom search engine (CSE) over the blogger search and normal google search is that you can determine how it indexes your site. If you submit your sitemap your pages will show up in the results. This makes search results from CSE more relevant than those from normal google search (even if you apply the site restriction to your site).
You can make money from adsense for search, in fact you can't opt out of ads if you don't want to pay for the professional account. I didn't want to have ads along with search results and I see this as a downside, however in the end the advantage of better results won me over.
If you want to see an example of a CSE, see the My Outsourced Brain custom search. You may also want to read an introductory article on how to set up a google custom search.
There are different choices on how to integrate the search results on your site in blogger. Some time ago I had the results embedded on my site in the sidebar, so they would always re-search for people coming from google or other search engines (similar to Chitika). During some efforts to make my site load faster I pulled the custom search result embedding because it required loading of several external files and because I got tired of all the customization efforts I put into making it look good.
Now I show the option which I currently use on this site, which is showing the results on a google site. Different to the code offered in the google custom search settings, the search form is standard HTML and does not show any google branding. The search bar is reset when you click on the search bar.
