"Apple is encouraging users to install and use XQuartz instead of X11," according to John Carter. He continues, "In fact, X11 is not included in Mountain Lion. This article explains.
"XQuartz is a community-supported version of the X11 windowing system for Mac OS-X 10.6.3 or later. Please visit http://xquartz.macosforge.org for more information.
"I have been using XQuartz instead of X11 in Snow Leopard and Lion for some time. However, I am unable to delete X11 (even on Lion) because it claims to be required by the OS; it seems apparent that they are sharing some of the same files. Having both installed has not been a problem. I don’t know if X11 will be automatically deleted when I upgrade to Mountain Lion.
"For anyone using GIMP, they must have either X11 or XQuartz installed. When XQuartz is running, the menu name is still X11, so unless you look for the version info you won’t know which one is running. Both X11 and XQuartz may be running at the same time, but launching one will not launch the other.
Here’s what XQuartz looks like in the Dock:

And here’s what X11 looks like in the Dock:

It is not necessary to run Xquartz (or X11) all the time, but once it is launched (it will launch automatically when GIMP is launched), it stays running until you manually quit X11.

When launching either X11 or XQuartz, a terminal window will also launch, but this terminal is an X11 version of Terminal.

Here’s the title bar for the X11 terminal window:

And here’s the title bar for the Terminal window:

If you have an X11 terminal window open, you really don’t need to also launch Terminal.
John's closing words, "You can close the X11 terminal window if you don’t want to see it, but you should not close XQuartz (or X11) if you have GIMP or some other application open that requires X11. When launching GIMP, the X11 terminal window will also open. Closing the X11 terminal window will not close GIMP."