Client is installed on each Mac that you want to view or control. You would usually just need to enable that in the Sharing prefs.
The control (ARD administrator) software is installed on the Mac that you want to use for, well, control of other Macs.
One example would be in a school computer lab, where all the student machines have ARD client enabled (through the Sharing preferences.) And, the teacher has a computer used to control all the student computers. That's an easy task for ARD, where the teacher can cause all systems to log out simultaneously, if needed.
And, the other difference is cost. Client is free, the ARD administration version is not.
Your first response appeared to show that you knew something about ARD, or had used it before - but, apparently not?