As part of the format process, you should be asked if you want to use MBR or GPT. That refers to the type of partition table.
You MUST initialize as GPT if you want to use more than 2.2 TB.
But I think you may have an insurmountable problem.
Your old hard drive is almost certainly MBR.
I'm not at all sure you can restore an image of an MBR disc to a GPT disc, which is what you would be doing.
A brief look at Google doesn't find any quick answer saying you can.
This may give you some more info:
http://www.sevenforums.com/backup-restore/345831-image-3tb-hdd-gpt-2t-image-mbr-possible.html
Are you willing to live with only 2.2 GB of that 3 TB drive? If you are, you should be able to do that. You'd just initialize the disk as MBR and forego the additional space.
But I'm guessing your new disc was initialized as MBR, so we don't know why you can't see the 2.2 GB.
So post the screen shot.
Did you in fact format this new disk at all? Or did you just install it as you bought it and then immediately restore that image? If you didn't format it, that may explain why you see only 500 GB.
Maybe someone else has more info about trying to restore an MBR image to a GPT disk. If it can't be done, you'd need to do a clean install instead of restoring that image.