The Custom Search Engine can also be fully integrated into a Google Site.Or you can Just Link your Custom Search Engine directly from the CSE URL address
Site Search is the Standard search for Google Sites. The Search Engine can be fully Integrated as explained here:
The Site search box is a standard feature of Google Sites, which provides a standard site search. A CSE Custom Search Engine can be integrated into the main header search box.
All the Search options are:
Google Sites' Site Search box is an integrated search for Google Sites.CSEs' Site Search is a paid business application designed for Custom Search Engine.
Custom Search Engine for Google Sites. The CSE Custom Search can be added to the main Google Sites search box. The CSE Search can provide many addition features to a search promotions, pagemaps, rich snippets, synonyms and many other refinements.
Custom Search Engine [Search Element] for CSE Embedding in Google Sites. The CSE Search Element can be added to a page as a part of a gadget.
The custom search is hosted independanly of any site so can be linked directly as a standalone web page.