Can you only use this on one computer?

i installed it on my laptop, it runs fine. my desktop runs better so i was going to play it there, it wont work. with the old key or a new one. why cant i just log in using my account info from my laptop? fuckin stupid.

You can use one account on multiple PCs. please refer How to backup & restore my account?. (This article is for backup & restore, but you can play the same on an old PC after restoring the new PC)