AOL is a different beast altogether than IE. AOL, as you’ve seen is one big window that contains a lot of smaller windows. If you close the big window, you’re disconnected from AOL itself. If you close a smaller window, you only close that window.
With IE, there’s only the one window - unless, of course, you open up multiple windows. To do this, you can click on FILE in the pull-down menus and then choose New >> Window. This will open up a whole new IE window that’s exactly the same as the window you’re using. Then you can close either of the two windows while still having IE itself open.