Just called Turbo Tax. The rep explained that when you purchase the physical disk, you can either install the disk OR download it using the code (not both!). Once you install the disk, it automatically uses that code and basically locks the code making it unable to be downloaded. You should be able to take that disk and install it in other computers with that disk, but you will be unable to download it and install WITHOUT the disk.
In the future, if you just use the download code (and do not install the disk), it will attach the license code to your intuit account and be accessible on additional computers as long as you are logged into that same account. It is ridiculous that Intuit does NOT make this clear on their software and packaging before you install the disk. The rep was very nice and obviously explained all of this to me, but unfortunately was unable to offer me a new code for the download on my non-disk-drive laptop.
