I don't know if I'm correct, but in my head, a window manager JUST manages windows. Gnome and KDE also manage windows, but they also contain applications for settings, printing, etc. Desktop Environments also have window managers, but they have more applications on top.

Window Managers manage windows as the name suggests and control how they are displayed and interacted with. A window manager is one component of a desktop environment which provides other facilities like compositors, task bars, status trays, task switchers, configuration applets, virtual desktops, and perhaps some default applications for basic things like terminal, file management, text editing, connection management, and image viewing. Some desktop environments feature extensive plug-in systems ( extensions ) and vast application ecosystems.

In the early days of Linux, there were no “desktop environments” and you would run a window manager directly over the window server ( eg. X11 ) with applications running directly over the WM. Proprietary UNIX introduced desktop environments like CDE, OpenWindows, and NeXTstep but, as they were proprietary, Linux lacked them. This changed with the advent of KDE and GNOME soon after. These days, the vast majority of Linux users are working with a desktop environment ( probably still one of these two though there are now others ).

A timing window manager in particular is a window manager that allows auto arranging and resizing applications to share the screen ( typically using keyboard commands ). The goal of a tiling window manager is that application views do not overlap and that the full desktop space is used efficiently. A floating window manager in contrast allows windows to overlap and leaves positioning, resizing, visibility, and focus up to the user. The desktop itself may be plainly visible and may even have clickable icons or applets displayed on it. Interaction with windows in a floating window manager is usually done with the mouse. Windows and Mac are examples of the floating metaphor so that is the one most of us are more familiar with. Any given window manager can incorporate both floating and tiling ideas and features but most WMs lean pretty heavily one way or the other.

Technically, a window manager is just a special kind of application. In X11, it is not even required. You can run applications directly without one but, if you run more than one application, you will quickly understand the value of a window manager. The value of a full desktop environment is more a matter of preference. Most people welcome them or consider them essential. Others see DEs as bloat. The middle ground is assembling a desktop experience yourself from a group of applications you select for that purpose from the window manager up.