You have it backwards. You can put your old drive into your new computer, but you will have to boot into the new operating system installed to your new hard drive and seek out the files you want to copy over to the new drive.
You will not be very successful booting the newer computer from the old drive and it's operating system for several reasons.
1) It's an OEM setup and that will inactivate windows
2) Even if it was Retail copy of sofware on the drive, the chips on the newer motherboard are not the same as on the older motherboard, so those driver files won't work properly.
3) At best you will boot the old hard drive to a crippled system, or it wont' boot at all.
I recently bought a Dell XPS 8700 running Win 10. I still have my old computer, a Dell Inspiron 530 running Win XP Home 5.1.2600. I made a clone (i.e., a bootable drive) of the old Inspiron 530 hard drive on a Seagate Expansion portable drive with a USB interface. I want to plug the Seagate drive into my new XPS 8700 and boot from the Seagate drive, thus starting up and running Win XP and accessing all my old apps and files. When I am done, I want to restart the XPS 8700 and start up Win 10 as usual. My XPS 8700 with Win 10 is running just fine. My purpose is simply to physically throw away my old computer and yet continue to use my old computer in a virtual sense by "hosting" the same functionality on my new computer. I realize I will need to restart my computer when moving from Win XP back to Win 10, and I realize I can't within one environment access files in the other environment. My immediate problem is I cannot get to a screen that lets me boot from my Seagate drive. (I might run into other problems after that.) I've spent hours looking online and in a Win 10 book, and can't find an answer. Thank you so very much for any help!