No. They are not the same. They are completely different, although closely related. Let me explain, for those of you who are unaware.
Before I get into the nitty-gritty details, let's all begin on a common ground: Windows. No doubt, you use Windows. I don't care if you have all of your computers switched over to Linux, Mac, UNX, VMS or anything else. Admit it. You use it. Whether it be at the library, school, your parents/kids/friends/relatives/etc. house, work or home, you use it. Heck- I'm using it to make this post. Windows is the most predominant operating system out there, and as such, hard to completely 100% ignore. So, because you use it, you are no doubt familiar with the interface. You have a Start menu, a taskbar, a system tray, a dekstop and probably some quick-launch icons. When you open a folder on your desktop, or anywhere else, you launch Windows Explorer to navigate the system and find what you need. You are probably familiar with other aspects of Windows, such as the Control Panel, My Computer, Network Places, the registry and other such functionality. Everything is combined into a nice smooth interface to keep the operating system user friendly.
With Linux, and most UNIX-like operating systems, it's quite a bit different. When you boot the computer, one of the very first things you launch is the kernel. Think of the kernel as the "workforce" of the operating system. It tells every application, system interal and configuration file how to behave. It manages I/O, interrupts and pretty much the entire OS. Once the kernel is booted, it tells the operating system that it is time to launch a graphical user interface (GUI). With Linux, it happens to be the X server. The X server is responsible for all the graphical representations on your monitor. It works seamlessly with your graphics engine, and you hardly know it exists. It is commonly referred to as just "X".
Once X is running, your GUI is ready to be drawn. However, all is not peaches and cream yet. We need a way to interact with the OS for the user. In walks the desktop manager. The desktop manager "manages" all the tasks that you do on your Linux system. In otherwords, moving a file from one folder to another is the responsibility of the desktop manager. Installing and uninstalling software, changing configuration files, playing games and launching software is all the responsibility of the desktop manager. In other words, it manages your desktop. The most common desktop managers for Linux are Gnome, KDE, XFCE and CDE.
Once the desktop manager is running, your windows manager can now be drawn. That's right, the windows manager is the graphical representation of the desktop manager. The buttons, themes, windows, menus, taskbars, etc. are all the responsibilty of the window manager. KDE, XFCE and CDE all have their very own window managers built right into the desktop manager. This is very similar to the way Windows behaves. Gnome, on the other hand, uses a seperate window manager called Metacity. This means that you can have a different window manager running with any desktop manager. For example, Ubuntu uses the Gnome desktop manager with the Metacity window manager. I can install any window manager I please with Gnome, and launch it when I login. Some of the common window managers are Matacity, Blackbox, Fluxbox, FVWM, OpenBox, Enlightenment, Afterstep and Sawfish. There are many others, some very outdated, some commercial. You could call Windows Explorer the window manager in Windows.
Now that you understand the difference between desktop managers and window managers, there is one final note. Regardless of the desktop manager and the window manager, any application can be open regardless. KDE uses Konqueror as it's GUI file manager. However, I can launch Konqueror in Gnome running the Blackbox window manager. In other words, there is not a window manager or desktop manager application that only works with that manager. It can be run in any DM or WM regardless (so long as it is installed).
This post came as I gained the knowledge from Kyral at UbuntuForums and feel obligated to share the info.