Thie module vs. Google Adsense module

Google Custom Search Engine (CSE) is an embedded search engine that can be used to search any set of one or more sites. No Google API key is required.

I think that explains well what the difference is. This module is not about Google Adsense.

So, Adsense provides both the ad and search functionality but Google CSE only does the search part.

That is correct.
I am not sure why a module for Google Adsense should provide also Google CSE, but that is what the other module does.

Confusingly, I believe that Google CSE module can also allow you to make money via AdSense, see http://www.google.com/coop/cse/ (although I've only used Google CSE module for a non-profit organization that didn't want ads so haven't tried this).

I know this has been closed for 7 years, but as the adsense maintainer, my input might be useful for anyone that finds this issue.

The reason why the adsense module also creates CSE forms and results is because "Search" (via CSE) is one of the AdSense options. From what I can see of the CSE code generated at cse.google.com, the adsense version is an extremely old version, that even precedes V1 of the CSE code (call it V0?).

I'm considering adding the CSE V2 code to the adsense module, as there seems to be issues with async JS and SSL support by the V0 code, but I'd much rather delegate that to this module. It does seem strange that the google_cse module doesn't add the partner-pub-nnnnn:yyyyy variable that is generated at cse.google.com when using the "Make money" option.

So yes, there is some overlap. But the adsense and google_cse modules mostly address different products provided by Google which share some common functionality.

When viewing the custom search engine in AdSense look for a link labeled Access Advanced Search Features. Under Setup this will show you the search engine ID. And I can confirm it does follow the partner-pub yada yada format.