The two are not the same thing at all, and likely you’ll use both.
The Administration Menu stuff shows you how to create the menu item and page for your settings to live on. In other words, the menu item in the left hand bar, and the existence of the page itself.
The Settings API is what you use to actually build the contents of that page (the table of settings, form fields, and submit button), and to then process the data coming back from it.