The initial file, whether a disk image or a zip or sit containing a disk image, has served its purpose. You can delete it.
The other icon is a mounted disk image. This is basically treated as a "virtual disk" that will stay mounted until you eject it or log out. That mounted image contains either an installer (which will need to be run, if you haven't already done so) or simply the application itself. If it contains the app, you need to copy the app to your applications folder (and voila! it's installed!).
Mounting a disk image does not "install" a application, it merely decompresses the file containing it. You usually have to install it yourself after that (as mentioned above).
It's really the easiest way to install applications I've ever seen.