A few things happen with an "open" program, like the ability to command-tab through them, appearing in the dock, expecting a window to be in a certain place, showing up in the "Force quit" dialog, etc.

If the user expects to to find an app in the dock, and it's not there, then the user does care if the application is open.

A feature like this doesn't matter as much on an iOS device because of the way applications are launched & managed. Applications are launched and managed differently on the desktop though...