Asynchronous JavaScript uses Promise objects to refactor code from callbacks, enabling robust, reusable, maintainable, and readable asynchronous functionality. The project refactors an existing program with multiple API calls using traditional callbacks into a structure based on promise chaining using the `.then` method. It demonstrates error handling with the `.catch` method and incorporates external modules into the asynchronous functions called in the promise chain. Promises can be chained together, include error rejection events, and errors are handled once throughout the chain. The refactored code is more maintainable and easier to debug than traditional callback structures.