But when I click on a button that is supposed to maximize the window to cover my whole screen
See, your premise is incorrect. The green button is not labeled "maximize" or "Full Screen". It is labeled "Zoom" (although Mac Help refers to it as the "Enlarge" button). It's not "supposed to maximize". Maybe it should, but it isn't mislabeled or misrepresented.
The green button works exactly the way its equivalent does in Mac OS Classic -
In the Finder, it toggles between the user setting and the setting that is necessary to show all of the window's contents.
In other apps, it toggles between the User Setting (the size you manually sized it to) and as large as will fit on the screen, leaving room for the Dock.
"Full Screen" is a completely different function, and some apps offer it and some don't. To me it's useless, because I want access to the Dock and the menu bar, which has to be somehow kludged in if an app takes over the entire screen. Then you are left wondering which arcane key combo makes the menu bar appear again so you can exit full-screen mode. The smarter apps just make the menu bar appear when you pass the cursor over the top edge, but I have seen others which require command-2 or some such nonsense.