Do you have an installation disk for windows or a repair disk if you created one when you first got machine?
This will be needed to attempt boot repair as the hardware in the different computers may make it not want to boot.
You could try creating an image backup of the computer HDD on to an external HDD. then putting the Hard drive from the new computer restoring the image to it choosing to reformat it as well when offered. Then it should boot OK.
I guess it comes down to how much work you want to do to have it boot from the old hard drive. Also the Microsoft tool Sandiver was talking about is SYSPREP this may also work to get your hard drive to boot correctly.