Why doesn't "exit" close a Terminal.app window on Mac OS X?
$ exit logout [Process completed]
Is there a way to close the window without using the mouse?
A window displayed by Terminal.app is just the frontend for the process you choose to run inside of it - in your case, a Unix shell. When you exit the shell, the Terminal.app does not close the window by default, so you have the possibility to inspect the output from whatever command you ran, after it finishes.
You can change your preferences here
Besides that, you can (almost) always close windows in OSX with Cmd-W, so you don't need mouse even if it doesn't close automatically.
One more hint: I like hitting Ctrl-D instead of typing exit. Two keys vs. five.
If you want to terminate the application itself from the commandline:
Command + Q -> closes the application/process.
Command + W -> closes window/instance