Module Builder allows you to easily add new functionality to SugarCRM though an easy to use GUI. If you want to track company Desks, Widgets, Recordings, Headaches,....anything, you would use Module Builder to add your specific needs to SugarCRM. You can even relate these new module items to the core modules such as contacts, accounts, etc.
Web Services API is a way to access SugarCRM data and functionality from non-SugarCRM code. Web Services API can be used to Create, Read, Update, and Modify existing SugarCRM records through API calls. You can use SOAP or REST to communicate with SugarCRM. This includes accessing any newly created custom modules you built using Module Builder. With Web Services API, you could access how many resources, Widgets, Recordings, or Headaches from another internal or external application.