You do NOT need to re-install, but you will need to re-activate because the license is tied to the motherboard. You can use this guide to use the same key with the new computer: [ Register or Signin to view external links. ]

Press Win+R to open the Run prompt, type "sysprep", and press enter.

Use these settings when you're ready to use the drive with the new computer.

[ Register or Signin to view external links. ]

This removes all of the drivers specific to your hardware so the OS can accept new hardware. When you boot to the drive in a new computer, Windows will automatically use the basic drivers it needs.